Let A, B, C and D be placed consecutively on a circle. Let W, X, Y and Z be the midpoints of the arcs AB, BC, CD and DA, respectively.

Question

Show that the chords WY and XZ are perpendicular.

I've drawn it using Geogebra and it is quite obvious that it is true - regardless of how I manipulate it, I just don't know where to start with proving it.

If I draw line segments WX and YZ I have two similar triangles, but that isn't enough to show that the two segments are perpendicular. Any ideas to get me in the right direction?

Quang Hoang · Accepted Answer · 2015-09-23T01:55:46.547

0

Let $I$ be the intersection of $WY$ and $XZ$. Then $$\angle WIX=\angle WZX+\angle ZWY.$$ Can you continue from here?

edited Sep 23 '15 at 01:55

answered Sep 23 '15 at 01:42

Quang Hoang

15,854

Do you mean ∠WIX=∠WZX+∠ZWI? – Charles Sep 23 '15 at 01:54
Yes, edited. Thanks. – Quang Hoang Sep 23 '15 at 01:56
I'm having a hard time seeing how you got that.. – Charles Sep 23 '15 at 02:20
They are both equal to $180^\circ-\angle WIZ$. – Quang Hoang Sep 23 '15 at 02:22
Nevermind, I was stuck on something else but I got it, all good now. – Charles Sep 23 '15 at 02:30

Let A, B, C and D be placed consecutively on a circle. Let W, X, Y and Z be the midpoints of the arcs AB, BC, CD and DA, respectively.

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