Problem 25
Let ABC be aright triangle, where B = 90°. Let AD be the altitude from A, where D lies on BC. If P, Q, and R are the incenters of triangles ΔABC, ΔADB, and ΔADC respectively, then show that:
AP = QR
Here’s a simulation that demonstrates the problem.
Try dragging the points B and C. Is AP always equal to QR? How can we prove the same?
Solution
To solve this problem, you’ll need to know the following.
Incenter of a Triangle
Cosine Rule
[coming soon]
Pythagoras Theorem
I’m assuming that you know this one. Anyways, here’s a quick recap:
In a right triangle, the square of the hypotenuse equals the sum of the squares of its other two sides.
Conversely:
If the square of one side of a triangle equals the sum of the squares of its other two sides, then this triangle must be right-angled.
Here’s a little simulation that demonstrates the theorem.
You can drag the three vertices of the right triangle. Is PQ2 always equal to QR2 + RP2?
Problem Source: MathClub
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