# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Perimeter To Angle? Topic: Geometry/Trigonometry. Level: Olympiad. April 16th, 2006

Problem: (1986 China TST – #5) Given a square whose side length is , and are points on the sides and , respectively. If the perimeter of is find the angle .

Solution: Let and . By the given condition, we have (1). From this, we find

Substituting from (1), we have

(2).

Note that and . Then

.

But by (2), we have .

Hence . QED.

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Comment: Trigonometry can come in handy quite often, especially when dealing with angles. Purely geometric solutions to this problem are a lot more complicated in my opinion; when in doubt, use algebra. The following identity comes in handy on quite a few problems.

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Practice Problem: Prove that in any triangle we have .

### 8 Responses to “Perimeter To Angle? Topic: Geometry/Trigonometry. Level: Olympiad.”

1. QC Says:

Inscribing a circle into triangle ABC, and using a = x + y, b = y + z, c = z + x, and r, we find that the identity is equivalent to

r^2 / xz + r^2 / xy + r^2 / yz = 1

By Heron’s, the area of the triangle A = \sqrt{xyz(x+y+z)} = (x+y+z)r, so A^2 / A^2 = 1 = r^2 (x + y + z) / xyz = 1, which is equivalent to the above identity. QED.

2. Anonymous Says:

Don’t you think you should start to slow down a little. USAMO is in two days!!!

Tomorrow is my cool down period for the year .

4. Anonymous Says:

Haha, nice. Well, good luck on the USAMO. You are a great role model. You’ve worked hard and improved so much in one year.

5. Anonymous Says:

I wonder if this proof is valid:

From the equations AP + AQ + PQ = 2, AP = 1 – PB, and AQ = 1 – QD we get
PQ = PB + QD

This means that triangles PCB and QCD can be “flipped” inward along PC and QC respectively to fill up triangle PCQ.

Then we have 2(alpha + beta) = 90, or alpha + beta = 45 which means alpha + beta = angle PCQ = 45.

That’s interesting. Intuitively, it makes sense, but I’m not completely sure if I would be able to rigorously prove it. A good idea, though!

7. Anonymous Says:

Hmm, I guess “flipping triangles” inward is not the best way to put it.