Geometry Problem 1573: Find the Angle at K. A High School and College Challenge

In the figure, two circles are externally tangent at point C. Line DE is the common external tangent, and secant FG passes through C. From point H, secants HIF and HGJ are drawn, with the angle at H measuring 56 degrees. The extended chords ID and JE intersect at point K. Find the angle at K.

Circles touch at C,
Secants and tangents align,
What angle does K see?

Key Definitions and Descriptions

Key Term Description
Circles tangent at C Two circles touch externally at a single point C.
Common external tangent DE A line that touches both circles at different points without crossing them.
Secant FG A line that intersects the circle at two points, passing through point C.
Secants HIF and HGJ Lines drawn from point H that intersect the circles at points I, F, G, and J.
Extended chords ID and JE Line segments ID and JE that intersect outside the circle at point K.
Angle at H The given angle at point H is 56 degrees.
Angle at K The unknown angle to be calculated where extended chords ID and JE intersect.