Geometry Problem 1580: 60-Degree Triangle Challenge: Find AD with Excenter and Circumcircle Clues! Perfect for High School and College Students!

In triangle ABC, angle B is 60 degrees, and D is the excenter relative to side BC. The extension of line DC meets the circumcircle at point E. If DE is 10 units, find AD.

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Hints, Key Definitions and Descriptions

Key Term	Definition
Triangle ABC	A triangle with vertices labeled A, B, and C.
Angle B	The angle formed at vertex B, given as 60 degrees.
Excenter (D)	A point located outside triangle ABC, serving as the center of the excircle opposite side BC.
Excircle	A circle outside triangle ABC, tangent to one of its sides and to the extensions of the other two sides.
Line DC	A line passing through excenter D and vertex C of triangle ABC.
Circumcircle	A circle that passes through all three vertices of triangle ABC.
Point E	The point where the extension of line DC intersects the circumcircle of triangle ABC.
Cyclic Quadrilateral	A quadrilateral whose vertices all lie on a single circle, called the circumcircle.
Segment DE	A line segment from point D (the excenter) to point E on the circumcircle, measuring 10 units.
Segment AD	The line segment from vertex A to excenter D, whose length needs to be determined.
Isosceles Triangle	A triangle with two sides of equal length.
Special Right Triangle 30-60 Degrees	A right triangle with angles measuring 30, 60, and 90 degrees, often featuring side lengths in the ratio 1:√3:2.
Auxiliary Line	A line added to a geometric figure to aid in solving a problem or proving a theorem.

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