A friend, Teddy O'Connell, sent me this geometry problem he found on the Web, which is not too hard and has a pretty neat solution.
Let ABC be a right triangle, with a right angle at C. Suppose that a circle has been inscribed in the triangle (the incircle) which is tangent to ABC at one point on each side including the point D on the hypotenuse AB, such that |AD| = 3 and |DB| = 5. Find (ABC), the area of triangle ABC.
(drawing not to scale)
I will follow this posting with two comments. The first will contain a couple of hints, and the second (to be posted later) is my solution and a related question that I don't have an answer to.
2 comments:
Hints for a solution:
(1) I think this problem is actually easier if you generalize it, assuming the given lengths are x and y rather than 3 and 5.
(2) All you need to know about the incircle is that any line segment drawn from the center of the circle (O) to a point on the circle (such as D) is a radius and so has the same length (usually denoted r) and is perpendicular to the line that is tangent to the circle at the end of the radius.
Easy algebra - lots cancels to give a nice answer which I won't post as a spoiler.
How about a synthetic solution that <1> explains<\i> the answer.
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