Problem 14
Let ABC be an isosceles triangle with AB = AC. Let P and Q be points on AB and AC respectively, such that AP = CQ. Show that the circle passing through A, P, and Q passes through the circumcenter of ABC.
Here’s a simulation that demonstrates the problem.
Here, O is the circumcenter of the triangle ABC. Try dragging the point P. The point Q will move such that AP = CQ. Does the circle passing through A, P, and Q always pass through O?
Solution
Here’s what you’ll need to know to solve this problem.
Cyclic Quadrilaterals
The sum of the opposite angles of a quadrilateral is always 180°. Conversely, if the sum of the opposite angles of a quadrilateral equals 180°, then the quadrilateral must be cyclic.
Here’s a simulation that demonstrates this property.
You can drag the vertices and observe the sum of the opposite angles. Is the sum always 180°?
Congruent Triangles
If the corresponding sides and angles of two triangles are equal, then they are said to be congruent.
And, if we’re able to prove that two triangles are congruent, then their corresponding parts (sides and angles) are equal.
There are various ways to prove the congruence of two triangles, one of them being the Angle-Side-Angle or ASA rule.
That is, if for two triangles ABC and PQR that AB = PQ, ∠A = ∠P, and ∠B = ∠Q, then the two triangles would be congruent.
And, this would mean that AC = PR, BC = QR, and ∠C = ∠R (i.e., the corresponding parts are equal).
You can observe this in the following simulation.
Drag the points A or B to make AB equal to PQ. Then, drag the two sliders to to make ∠A and ∠B equal to ∠P and ∠Q respectively. Then, is ∠C equal to ∠Q, AC equal to PR, and BC equal to QR?
Here’s, how you can proceed now.
- To prove that the circle through A, P, and Q passes through O, you need to prove that APOQ is a cyclic quadrilateral.
- For that, you need to prove that ∠PAQ + ∠POQ = 180°.
- And, for proving this sum to be 180°, you’ll first need to find a pair of congruent triangles. Then, you can use the equality of their corresponding angles to proceed.
That’s all from my side. Good luck!
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
Problem Source: CutTheKnot
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