Really Hard Circle Problem

Question

Chords AB and CD cross at right angles at point P. If AP=3, BP=8, and CP=10, compute the radius of the circle.

Answer 1

Per comments above you can determine a circumcenter for the circle passing through points A, B, C, then determine the distance from circumcenter to any of those points (including D which better lie on the circle too).

Using Law of Cosines you can determine measure of an inscribed angle using arctan(10/3) (for example), then the measure of a central angle intercepting the same arc will be twice that. You can determine the length of the segment intercepting that arc using Pythagorean Theorem. Now you have an isosceles triangle with a known vertex angle, the base side, and the sides including the vertex angle are equal (both radii). This enables you to use the Law of Cosines.

You could also determine the measure of a base angle of the isosceles triangle, then use the Law of Sines to determine the radius.

This graph shows that all three techniques lead to the same value for the radius of the circle.

desmos.com/calculator/lejfdnovt4

Enable each of the folders in turn.