Answer to Question #88472 in Geometry for Zyrelle

Question

Given:<ABC=<CDA,AP=CP
Prove:arc AB=arc CD

Answer 1

Since <ABC=<CDA and inscribed angles means the CDA arc is equal to the ABC arc. Therefore as the arc CDA is equal to the arc arc ABC and CDA + arc ABC = 360 degrees so the arc CDA = arc ABC = 180 degrees and <ABC=<CDA =90 degrees. Therefore, ABCD is a rectangle, then AB=CD.

Let the point O be the center of the circle. Then the angle of AOB and COD are the central angles. If the angle AOB = COD then the arcs on which they are based are also equal, i.e. arc AB =arc CD