Im working to prove that an interior point on a chord is also interior to the circle containing the chord. Ive got the entire proof almost finished and laid out, however im stuck on one part. My proof would be complete if I could show this: Given an isoceles triangle, in absolute geometry (no parallel postulate, sum of triangle <= 180) For an arbitrary point P, interior to segment AC. Prove that segment PB is always less then AB=CB. I establish that since P is interior to AC, P will never equal A or C. From there Ive been trying to establish that angle A or C will always be smaller then the angle created by P, so that by the scalene inequality side AC or BC will always be larger (since they correspond to angles created by P). However I cannot figure out how to make a general proof for this in absolute geometry, can anyone offer any insight?