Problem 4
Four points A, B, C, and D are taken on a circle. If P, Q, R, and S are the midpoints of arcs AB, BC, CD, and DA respectively, show that:
PR ⊥ QS
Here’s a simulation that demonstrates what needs to be proven.
You can drag the points A, B, C, and D (keeping them in cyclic order) and observe the angle between PR and QS. Is it always a right angle?
Solution
I won’t provide the full solution here, but only give you some tools and hints that’ll help you solve the problem yourself.
Three things to know here.
First, the angles of a quadrilateral add up to 360°. Let’s observe this in the applet below.
You can drag the four vertices and obseve the sum of the four angles. Is the sum always 360°?
Second, if two arcs of a circle are equal in length, then they subtend equal angles at the center of the circle. Let’s observe this in the applet below.
In the applet above, arc(PQ) = arc(PR). You can drag P and R and observe the angles that the arcs PQ and QR subtend at the center. Are they always equal?
Three, the angle subtended by a circular arc at the center of the circle is twice the angle subtended by it at the circumference. Let’s observe this in the applet below.
You can drag the points P, Q, and R, and observe the angles POQ and PRQ. Is ∠POQ always twice of ∠PRQ?
Well, that’s all that you need to know here. I suggest you proceed on your own from this point. 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 Facebook page.
Problem Source: CutTheKnot
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