Find the Radius of the Circle in This Right Triangle | Geometry Problem Solved

A full GeoMethod walk-through of a right-triangle circle-radius problem using the tangent-secant theorem, parallel lines, and the Pythagorean theorem. We build the key auxiliary structure step by step, transfer lengths carefully, and finish with the final algebra. 00:00 Problem: Find the circle radius r 00:06 Given: AB = AD = 3, CD = 1, CE tangent 01:08 Tangent-secant theorem: CE² = CD · CA 02:21 Solve: CE = 2 02:38 Pythagoras in △ABC: BC = √7 03:18 BE = BC − CE = √7 − 2 04:01 OE ⟂ CE and AB ⟂ CE ⇒ OE ∥ AB 05:26 Rectangle OFBE: OF = BE and BF = OE = r 07:07 Pythagoras in △AFO 08:09 6r = 20 − 4√7 ⇒ r = (10 − 2√7)/3 08:54 Theorem chain: tangent-secant → Pythagoras → rectangle 09:55 Result: r = (10 − 2√7)/3 Keywords: right triangle circle radius, find the radius geometry problem, tangent secant theorem, pythagorean theorem geometry, parallel lines geometry proof, circle geometry problem, right triangle proof, geometry problem solved, math olympiad style geometry, GeoMethod geometry