Centers Geometry

Question

Segments GH, HI, and IG are tangent to the circle with center C as shown. Given that ∠GHI = 70° and ∠GIH = 50°, find ∠HCI

Answer 1

There is a circle theorem that states that a line drawn from an external point (such as points G, H, and I) to the center of a circle (C) bisects the angle between the two tangent lines drawn from that point.

1. Hence HC bisects angle GHI, so m∠CHI = (1/2) m∠GHI
2. IC bisects angle GIH, so m∠CIH = (1/2) m∠CIH
3. For triangle HCI then, we must have m∠HCI = 180° - (m∠GHI + m∠CIH)