Geometry Problem 1617: Tangency, Concyclic Points, and Metric Relations in a Triangle

Geometry Problem 1617: Triangle ABC with concyclic points B, D, E, F, G, H and tangent circumcircle ADE

Figure: Intersection of the tangent line EH and the chord HF with side BC at point J.

Problem Statement:

Let ABC be a triangle. Let D be a point on side AB, and let E, F be points on side AC. Let G be a point on side BC. Suppose there exists a circle ω passing through the points B, D, E, F, and G.

Let ω₁ be the circumcircle of triangle ADE. The line tangent to ω₁ at E meets ω at a point H on the arc BG. Let the line HF intersect the side BC at point J.

Given:

GJ = 3
CG = 2

Find: The length of the segment FJ.

Join the Challenge!

Do you have a creative synthetic solution? We invite you to share your diagrams and reasoning with our community on the GoGeometry Blogger.

Post your solution on Blogger

Strategic Hints:

• Angle Relationships: Consider how the tangency at E connects the two circles through common points.
• Geometric Invariants: Look for equal arcs or directed segments resulting from the cyclic configuration.
• Similarity Search: Identify triangles sharing a vertex on line BC that satisfy an AA condition.
• Metric Discovery: Establish a ratio involving FJ by leveraging the identified similarity.