The students are currently learning about Mendelian genetics. They have a "packet" of four worksheets to fill out. The first page has a passage to read and the second page asks the students to take notes on the passage in outline form, which confuses the students for some reason. With the pure period rotation at this school starting with second, I must write down many names on the bad list.
I finally crack down on outline failures in fifth period. This time, I write down the steps for filling out an outline on the board. I also find another excuse to sing a song as an incentive to complete the packet, the "Packet Song." Actually, I'm changing it to "Packet Rap" as this song actually sounds better as a rap. I call it "packet" even though they ultimately them into interactive notebooks.
The only other change I made to the song is "four-page packet" instead of "ten-page packet," since the students have only four pages to work on today. The outline hints and rap work for the most part, since I only need to write one name on the bad list in fifth period and none in sixth -- after I coax one girl to finish enough questions to earn the reward for the class.
Dialogue 15 of Douglas Hofstadter's Godel, Escher, Bach is called "Birthday Cantatatata ..." Here's how the Dialogue begins:
"One fine May day, the Tortoise and Achilles meet, wandering in the woods. The latter, all decked out handsomely, is doing a jiggish sort of thing to a tune which he himself is humming. On his vest he is wearing a great big button with the words 'Today is my Birthday!'"
Tortoise: Hello there, Achilles. What makes you so joyful today? Is it your birthday, by any chance?
Achilles: Yes, yes! Yes it is, today is my birthday!
Tortoise: That is what I had suspected, on account of that button which you are wearing, and also because unless I am mistaken, you are singing a tune from a Birthday Cantata by Bach, one written in 1727 for the fifty-seventh birthday of Augustus, King of Saxony.
Achilles: You're right. And Augustus' birthday coincides with mine, so THIS Birthday Cantata has double meaning. However, I shan't tell you my age.
Well, Zeno first wrote about Achilles over 2500 years ago, so I assume that Achilles must be celebrating his 2500th birthday or more. Then again, Homer described Achilles several centuries before that, which would make the Greek hero's age closer to 3000.
Meanwhile, considering that it's almost May, I was considering delaying this chapter until his actual birthday that's coming up soon. But the king who shares his special day with Achilles was born on May 12th -- and unfortunately, that's a Sunday this year, hence a non-posting day. (Indeed, Sunday, May 12th is Mother's Day this year.)
Let's proceed a few lines in the Dialogue. In the Tortoise's usual fashion, he doesn't conclude it's truly his friend's birthday unless Achilles answers many more questions. This is just as we saw back in the Lewis Carroll Dialogue -- just because A and B are true, the Tortoise doesn't necessarily accept that Z is true. Here Z is "Today is the birthday of Achilles":
Tortoise: Why, who would ever have thought you to be gullible? Quite to the contrary, I regard you as an authority in the science of valid deductions, a fount of knowledge about correct methods of reasoning ... To tell the truth, Achilles, you are, in my opinion, a veritable titan in the art of rational cogitation. And it is only for that reason that I would ask you, "Do the foregoing sentences present enough evidence that I should conclude without further puzzlement that today is your birthday?"
So Achilles suddenly decides to answer "yes" infinitely many times. The reptile refers to this as "Answer Schema omega." Yet the Tortoise still demands one more thing before he finally accepts that it's his friend's birthday:
Achilles: What? That I ask for no present?
Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine birthday dinner, provided merely that I am convinced that knowledge of all those yes answers at once (as supplied by Answer Schema omega) allows me to proceed directly and without any further detours to the conclusion that today is your birthday. That's the case, isn't it?
Achilles: Yes, of course it is.
But the Tortoise just labels this as Answer Schema omega + 1 and then continues. Subsequent answers from Achilles are labeled omega + 2, omega + 3, omega + omega, omega + omega + omega, and finally omega * omega, which we can write as omega^2.
Tortoise: Well, after Answer Schema omega^2 there's answer omega^2 + 1. And then answer omega^2 + 2. And so forth. But you can wrap those all together into a packet, being Answer Schema omega^2 + omega. And then there are quite a few other answer-packets, such as omega^2 + 2omega, and omega^2 + 3omega ... Eventually, you come to Answer Schema 2omega^2, and after a whie, Answer Schemas 3omega^2 and 4omega^2. Beyond them there are yet further Answer Schemas, such as omega^3, omega^4. omega^5, and so on. It goes on quite a ways, you know.
Achilles realizes that subsequent answers can be labeled omega^omega, omega^omega^omega, and so on. The Tortoise tells him that these can be covered by Answer Schema epsilon_0. But then, not only does the Tortoise still not accept that it's his friend's birthday, the reptile reveals that it's actually his uncle's birthday. Thus he, the Tortoise, deserves a present, even though Achilles is the one whose birthday it really is:
Tortoise: Ah, but you never did succeed in convincing me of the veracity of that remark. You kept on beating around the bush with answers, Answer Schemas, and whatnot. All I wanted to know was if it was your birthday or not, but you managed to befuddle me entirely. Oh, well, too bad. In any case, I'll be happy to let you treat me to a birthday dinner this evening.
Achilles: Very well. I know just the place. They have a variety of delicious soups. And I know exactly what kind we should have ...
Chapter 15 of Douglas Hofstadter's Godel, Escher, Bach is called "Jumping out of the System." Here is how it begins:
"One of the things which a thoughtful critic of Godel's proof might do would be to examine its generality. Such a critic might, for example, suspect that Godel has just cleverly taken advantage of a hidden defect in one particular formal system, TNT."
The author reminds us that TNT is incomplete because G is true yet unprovable:
"I Cannot Be Proven in Formal System TNT"
or, expanded a bit,
"There does not exist a natural number which forms a TNT-proof-pair with the Godel number of this string."
So now Hofstadter asks, if we add G to TNT, is the resulting system TNT+G still incomplete?
"Since Godel's proof relies primarily on the expressive power of a formal system, we should not be surprised to see our new system succumb, too. The trick would be to find a string which expresses the statement:"
"I Cannot Be Proven in Formal System TNT+G."
Once again, the author likes to return to the MIU-system. We know that MU is unprovable in MIU, and so we can add MU to form the MIU+MU system. Here's a proof in MIU+MU:
Statements Reasons
1. MU 1. axiom
2. MUU 2. rule 2
Thus m = 30300, n = 300 is a MIU+MU-proof-pair, though not a MIU-proof-pair.
So Hofstadter returns to TNT and extends it to TNT+G. He begins by defining the formula:
(TNT+G)-PROOF-PAIR{a,a'}
and then we give it an "uncle":
~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(a",a')>
assign the uncle the Godel number u', and then arithmoquine it:
~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(u'S0/a",a')>
Tortoise: Why, who would ever have thought you to be gullible? Quite to the contrary, I regard you as an authority in the science of valid deductions, a fount of knowledge about correct methods of reasoning ... To tell the truth, Achilles, you are, in my opinion, a veritable titan in the art of rational cogitation. And it is only for that reason that I would ask you, "Do the foregoing sentences present enough evidence that I should conclude without further puzzlement that today is your birthday?"
So Achilles suddenly decides to answer "yes" infinitely many times. The reptile refers to this as "Answer Schema omega." Yet the Tortoise still demands one more thing before he finally accepts that it's his friend's birthday:
Achilles: What? That I ask for no present?
Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine birthday dinner, provided merely that I am convinced that knowledge of all those yes answers at once (as supplied by Answer Schema omega) allows me to proceed directly and without any further detours to the conclusion that today is your birthday. That's the case, isn't it?
Achilles: Yes, of course it is.
But the Tortoise just labels this as Answer Schema omega + 1 and then continues. Subsequent answers from Achilles are labeled omega + 2, omega + 3, omega + omega, omega + omega + omega, and finally omega * omega, which we can write as omega^2.
Tortoise: Well, after Answer Schema omega^2 there's answer omega^2 + 1. And then answer omega^2 + 2. And so forth. But you can wrap those all together into a packet, being Answer Schema omega^2 + omega. And then there are quite a few other answer-packets, such as omega^2 + 2omega, and omega^2 + 3omega ... Eventually, you come to Answer Schema 2omega^2, and after a whie, Answer Schemas 3omega^2 and 4omega^2. Beyond them there are yet further Answer Schemas, such as omega^3, omega^4. omega^5, and so on. It goes on quite a ways, you know.
Achilles realizes that subsequent answers can be labeled omega^omega, omega^omega^omega, and so on. The Tortoise tells him that these can be covered by Answer Schema epsilon_0. But then, not only does the Tortoise still not accept that it's his friend's birthday, the reptile reveals that it's actually his uncle's birthday. Thus he, the Tortoise, deserves a present, even though Achilles is the one whose birthday it really is:
Tortoise: Ah, but you never did succeed in convincing me of the veracity of that remark. You kept on beating around the bush with answers, Answer Schemas, and whatnot. All I wanted to know was if it was your birthday or not, but you managed to befuddle me entirely. Oh, well, too bad. In any case, I'll be happy to let you treat me to a birthday dinner this evening.
Achilles: Very well. I know just the place. They have a variety of delicious soups. And I know exactly what kind we should have ...
Chapter 15 of Douglas Hofstadter's Godel, Escher, Bach is called "Jumping out of the System." Here is how it begins:
"One of the things which a thoughtful critic of Godel's proof might do would be to examine its generality. Such a critic might, for example, suspect that Godel has just cleverly taken advantage of a hidden defect in one particular formal system, TNT."
The author reminds us that TNT is incomplete because G is true yet unprovable:
"I Cannot Be Proven in Formal System TNT"
or, expanded a bit,
"There does not exist a natural number which forms a TNT-proof-pair with the Godel number of this string."
So now Hofstadter asks, if we add G to TNT, is the resulting system TNT+G still incomplete?
"Since Godel's proof relies primarily on the expressive power of a formal system, we should not be surprised to see our new system succumb, too. The trick would be to find a string which expresses the statement:"
"I Cannot Be Proven in Formal System TNT+G."
Once again, the author likes to return to the MIU-system. We know that MU is unprovable in MIU, and so we can add MU to form the MIU+MU system. Here's a proof in MIU+MU:
Statements Reasons
1. MU 1. axiom
2. MUU 2. rule 2
Thus m = 30300, n = 300 is a MIU+MU-proof-pair, though not a MIU-proof-pair.
So Hofstadter returns to TNT and extends it to TNT+G. He begins by defining the formula:
(TNT+G)-PROOF-PAIR{a,a'}
and then we give it an "uncle":
~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(a",a')>
assign the uncle the Godel number u', and then arithmoquine it:
~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(u'S0/a",a')>
Its interpretation is
"There is no number a that forms a TNT+G-proof-pair with the arithmoquinification of u'."
More concisely,
"I Cannot Be Proven in Formal System TNT+G."
And of course there's no point in calling this new Godel statement G' and adding it to TNT+G, since TNT+G+G' will be just as incomplete. In fact, the author explains that we can add Godel statements to TNT+~G and its descendants as well:
"Each time we make a new extension of TNT, its vulnerability to the Tortoise's method -- pardon me, I mean Godel's method -- allows a new string to be devised, having the interpretation:"
"I Cannot Be Proven in Formal System X."
And he asks, what if we try to add all the possible G's to TNT at once, as in TNT+G_omega? Once again, we return to the Tortoise in the Dialogue, and define a new symbol:
(TNT+G_omega)-PROOF-PAIR{a,a'}
Arithmoquine its uncle, and voila -- we have another new undecidable uncle. Hofstadter asks, why wasn't G_omega already included as one of the added G's?
"The answer is that G_omega was not clever enough to foresee its own embeddability inside number theory. In the Contracrostipunctus, one of the essential steps in the Tortoise's making an 'unplayable record' was to get a hold of a manufacturers blueprint of the record play which he was out to destroy."
The author now returns to Cantor's diagonal argument along with a very commonly given objection -- why can't we just add the diagonal number to the list?
(1a) Take list L, and construct its diagonal number d.
(1b) Throw d somewhere into list L, making a new list L+d.
(2a) Take list L+d, and construct its diagonal number d'.
(2b) Throw d' somewhere into list L+d, making a new list L+d+d'.
...
But even if we add all of the d numbers into the list somehow, we'd just obtain a new list which we can diagonalize yet again. The same is true of TNT:
"TNT is therefore said to suffer from essential incompleteness because the incompleteness here is part and parcel of TNT; it is an essential part of the nature of TNT and cannot be eradicated in any way, whether simple-minded or ingenious."
The author tells us that there are three basic conditions that must be satisfied in order for there to be an undecidable string (or an unplayable record):
(1) That the system should be rich enough so that all desired statements about numbers, whether true or false, can be expressed in it. (Otherwise you have a refrigerator, not a phonograph.)
(2) That all the general recursive relations should be represented by formulas in the system. (Or otherwise you have a record player of low fidelity.)
(3) That the axioms and typographical patterns defined by its rules be recognizable by some terminating decision procedure. (Otherwise you have a partially-designed phonograph.)
"The threshold seems to be roughly when a system attains the three properties listed above. Once this ability for self-reference is attained, the system has a hole which is tailor-made for itself; the hole takes the features of the system into account and uses them against the system."
Now Hofstadter quotes author J.R. Lucas, who attempts to use a Godelian argument to show that Artificial Intelligence can never reproduce human intelligence -- that is, that Turing's test will always fail and the singularity will never occur. Here is the gist of it:
Rigid internal codes entirely rule computers and robots; ergo ...
Computers are isomorphic to formal systems. Now ...
Any computer which wants to be as smart as we are has got to be able to do number theory as well as we can, so ...
Among other things, it has to be able to do primitive recursive arithmetic. But for this very reason ...
It is vulnerable to the Godelian "hook," which implies that ...
We, with our human intelligence, can concoct a certain statement of number theory which is true, but the computer is blind to that statement's truth (i.e., will never print out), precisely because of Godel's boomeranging argument.
This implies that there is one thing which computers just cannot be programmed to do, but which we can do. So we are smarter.
Before Hofstadter attempts to rebut Lucas here, he shows us an Escher drawing -- Dragon. It's all about a dragon biting its own tail. Now the author quotes the artist:
I. Our three-dimensional space is the only true reality we know.
II. However much this dragon tries to be spatial, he remains completely flat. Two incisions are made in the paper on which he is printed. Then it is folded in such as way as to leave two square openings. But this dragon is an obstinate beast, and in spite of his two dimensions he persists in assuming that he has three; so he sticks his head through one of the holes and his tail through the other.
Hoftstadter explains:
"But the dragon cannot step out of his two-dimensional space, and cannot know it as we do. We could, in fact, carry the Escher picture any number of steps further."
And indeed:
"Each time at the instant that it becomes two-dimensional -- no matter how cleverly we seem to have simulated three dimensions inside two -- it becomes vulnerable to being cut and folded again. Now with this wonderful Escherian metaphor, let us return to the program versus the human."
He tells us that just as computers have a limit as revealed when the human Godelizes, so do humans have a limit -- we can't Godelize at will. Just ask Achilles. The author returns to the birthday story, where the Tortoise runs out of names such as omega, omega^omega, omega^omega^omega:
"Thus a new name must be supplied ad hoc. Now offhand you might think that these irregularities in the progression from ordinal to ordinal (as these names of infinity are called) could be handled by a computer program."
But as mathematicians Alonzo Church and Stephen C. Kleene proved, it can't:
There is no recursively related notion-system which gives a name to every constructive ordinal.
Speaking of ordinals and their names, I've discussed ordinals on the blog before. It was in the context of the Common Core Standards. Recall that there was much ado concerning whether:
5 * 3 = 3 + 3 + 3 + 3 + 3
or
5 * 3 = 5 + 5 + 5
The first method is recommended by some Common Core texts. But the second method is actually consistent with how ordinals are defined, because:
omega * 3 = omega + omega + omega
Ordinal multiplication is not commutative. In fact:
3 * omega = 3 + 3 + 3 + 3 + ...
= omega
since the limit of the natural numbers 3, 3 + 3, 3 + 3 + 3, is just omega. (In terms of Achilles and his birthday, if he said "yes yes yes" three times at each step, then he skips to steps 3, 6, 9, ..., which are still all covered by Answer Schema omega.)
But unfortunately, Hofstadter in his Dialogue gives omega + omega the name 2omega. This is not correct -- omega * 2 is omega + omega, but 2omega (or 2 * omega) is just omega.
Indeed, we don't even need infinite ordinals to refute Hofstadter and the Common Core -- Axiom 5 of TNT states that:
Axiom 5: Aa: Ab: (a * Sb) = ((a * b) + a)
So we write:
Statements
1. Aa: Ab: (a * Sb) = ((a * b) + a) [axiom 5]
2. (SSSSS0 * SSS0) = ((SSSSS0 * SS0) + SSSSS0
3. (SSSSS0 * SS0) = ((SSSSS0 * S0) + SSSSS0
4. (SSSSS0 * SSS0) = (((SSSSS0 * SS0) + SSSSS0) + SSSSS0
5. (SSSSS0 * S0) = (SSSSS0 * 0) + SSSSS0
6. (SSSSS0 * SSS0) = ((((SSSSS0 * 0) + SSSSS0) + SSSSS0) + SSSSS0)
7. (SSSSS0 * 0) = 0 [axiom 4]
8. (SSSSS0 * SSS0) = (((0 + SSSSS0) + SSSSS0) + SSSSS0)
which states that 5 * 3 = 0 + 5 + 5 + 5.
But let's get back to refuting Lucas. Humans are limited by what they can decide -- after all, we can't decide the Epimenides paradox:
"This was cleverly pointed out by C.H. Whitely, when he proposed the sentence 'Lucas cannot consistently assert this sentence.'"
The author asks, is it possible for either a human or a computer to be self-transcendent? For humans, it might seem to be possible:
"Similarly, it is entirely conceivable that a partial ability to 'step outside of itself' could be embodied in a computer program. However, it is important to see the distinction between perceiving oneself, and transcending oneself."
Now Hofstadter makes one last reference -- Jaich's four Dialogues in Are Quanta Real? which is based on a work by Galileo. There are three characters in these Dialogues:
"Why wouldn't two have sufficed: Simplicio, the educated simpleton, and Salviati, the knowledgeable thinker?"
Well, Sagredo, the third character, is a neutral third party, yet also helps prove Simplicio wrong.
Hofstadter concludes the chapter as follows:
"Somewhere along this elusive path may come enlightenment. In any case (as I see it), the hope is that by gradually deepening one's self-awareness, by gradually widening the scope of 'the system,' one will in the end come to a feeling of being at one with the entire universe."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The perimeter of rectangle ABCD is 32. Its length is 2 less than twice its width. If point E can be located anywhere along sideAB between points A and B, then what is the area of Triangle DEC?
We can use algebra to find the length and width of the rectangle:
w + 2w - 2 + w + 2w - 2 = 32
6w - 4 = 32
6w = 36
w = 6
2w - 2 = 10
Thus the area of the rectangle is 60. Notice that the area of the triangle is half that of the rectangle (no matter where E is), which is 30. Therefore the desired area is 30 square units -- and of course, today's date is the thirtieth.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The perimeter of rectangle ABCD is 32. Its length is 2 less than twice its width. If point E can be located anywhere along side
We can use algebra to find the length and width of the rectangle:
w + 2w - 2 + w + 2w - 2 = 32
6w - 4 = 32
6w = 36
w = 6
2w - 2 = 10
Thus the area of the rectangle is 60. Notice that the area of the triangle is half that of the rectangle (no matter where E is), which is 30. Therefore the desired area is 30 square units -- and of course, today's date is the thirtieth.
Lesson 15-6 of the U of Chicago text is called "Angles Formed by Tangents." In the modern Third Edition of the text, angles formed by tangents appear in Lesson 14-5.
This is what I wrote last year about today's lesson:
The theorems in this lesson are similar to those in yesterday's lesson.
Tangent-Chord Theorem:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Given:AB chord of Circle O, Line BC tangent to Circle O
Prove: Angle ABC = Arc AB/2
Proof:
Statements Reasons
1. Draw diameterBD. 1. Through any two points there is exactly one line.
2. Arc AD = 180 - AB 2. Arc Addition Postulate
3. CB perpendicularBD 3. Radius-Tangent Theorem
4. Angle ABC = 90 - ABD 4. Angle Addition Postulate
5. Angle ABC = 180/2 - Arc AD/2 5. Inscribed Angle Theorem
6. Angle ABC = (180 - Arc AD)/2 6. Distributive Property
7. Angle ABC = Arc AB/2 7. Substitution Property of Equality
Tangent-Secant Theorem:
The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.
Given: Line AB secant, Ray EC tangent at point C, forming Angle E,
Arc AC = x, Arc BC = y
Prove: Angle E = (x - y)/2
Proof ("between a tangent and a secant"):
Statements Reasons
1. DrawAC. 1. Through any two points there is exactly one line.
2. Angle DCA = x/2, EAC = y/2 2. Inscribed Angle Theorem
3. Angle DCA = EAC + E 3. Exterior Angle Theorem
4. Angle E = DCA - EAC 4. Subtraction Property of Equality
5. Angle E = x/2 - y/2 5. Substitution Property of Equality
6. Angle E = (x - y)/2 6. Distributive Property
In the text, the "between two tangents" is given as an exercise. The Given part of this proof with the way the points are labeled is completely different from the first part.
Given: Ray PV tangent at Q, Ray PU tangent at R
S on Circle O (same side ofQR as P), T on Circle O (opposite side of QR as P)
Prove: Angle P = (Arc QTR - QSR)/2
Proof ("between two tangents"):
Statements Reasons
1. DrawQR. 1. Through any two points there is exactly one line.
2. Angle VQR = Arc QTR/2, 2. Inscribed Angle Theorem
Angle PQR = Arc QSR/2
3. Angle VQR = PQR + P 3. Exterior Angle Theorem
4. Angle P = VQR - PQR 4. Subtraction Property of Equality
5. Angle P = Arc QTR/2 - QSR/2 5. Substitution Property of Equality
6. Angle P = (Arc QTR - QSR)/2 6. Distributive Property
In some ways, the Tangent-Chord Theorem is just like yesterday's Angle-Chord Theorem, except that one of the intercepted arcs is 0 degrees. The bonus question concerns a solar eclipse.
The theorems in this lesson are similar to those in yesterday's lesson.
Tangent-Chord Theorem:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Given:
Prove: Angle ABC = Arc AB/2
Proof:
Statements Reasons
1. Draw diameter
2. Arc AD = 180 - AB 2. Arc Addition Postulate
3. CB perpendicular
4. Angle ABC = 90 - ABD 4. Angle Addition Postulate
5. Angle ABC = 180/2 - Arc AD/2 5. Inscribed Angle Theorem
6. Angle ABC = (180 - Arc AD)/2 6. Distributive Property
7. Angle ABC = Arc AB/2 7. Substitution Property of Equality
Tangent-Secant Theorem:
The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.
Given: Line AB secant, Ray EC tangent at point C, forming Angle E,
Arc AC = x, Arc BC = y
Prove: Angle E = (x - y)/2
Proof ("between a tangent and a secant"):
Statements Reasons
1. Draw
2. Angle DCA = x/2, EAC = y/2 2. Inscribed Angle Theorem
3. Angle DCA = EAC + E 3. Exterior Angle Theorem
4. Angle E = DCA - EAC 4. Subtraction Property of Equality
5. Angle E = x/2 - y/2 5. Substitution Property of Equality
6. Angle E = (x - y)/2 6. Distributive Property
In the text, the "between two tangents" is given as an exercise. The Given part of this proof with the way the points are labeled is completely different from the first part.
Given: Ray PV tangent at Q, Ray PU tangent at R
S on Circle O (same side of
Prove: Angle P = (Arc QTR - QSR)/2
Proof ("between two tangents"):
Statements Reasons
1. Draw
2. Angle VQR = Arc QTR/2, 2. Inscribed Angle Theorem
Angle PQR = Arc QSR/2
3. Angle VQR = PQR + P 3. Exterior Angle Theorem
4. Angle P = VQR - PQR 4. Subtraction Property of Equality
5. Angle P = Arc QTR/2 - QSR/2 5. Substitution Property of Equality
6. Angle P = (Arc QTR - QSR)/2 6. Distributive Property
In some ways, the Tangent-Chord Theorem is just like yesterday's Angle-Chord Theorem, except that one of the intercepted arcs is 0 degrees. The bonus question concerns a solar eclipse.
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