# Difference between revisions of "Concurrence/Problems"

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==Intermediate== | ==Intermediate== | ||

+ | :''See []1992 AIME Problems/Problem 14]]'' | ||

+ | |||

==Olympiad== | ==Olympiad== | ||

Hallie is teaching geometry to Warren. She tells him that the three medians, the three angle bisectors, and the three altitudes of a triangle each meet at a point (the centroid, incenter, and orthocenter respectively). Warren gets a little confused and draws a certain triangle ABC along with the median from vertex A, the altitude from vertex B, and the angle bisector from vertex C. Hallie is surprised to see that the three segments meet at a point anyway! She notices that all three sides measure an integer number of inches, that the side lengths are all distinct, and that the side across from vertex C is 13 inches in length. How long are the other two sides? | Hallie is teaching geometry to Warren. She tells him that the three medians, the three angle bisectors, and the three altitudes of a triangle each meet at a point (the centroid, incenter, and orthocenter respectively). Warren gets a little confused and draws a certain triangle ABC along with the median from vertex A, the altitude from vertex B, and the angle bisector from vertex C. Hallie is surprised to see that the three segments meet at a point anyway! She notices that all three sides measure an integer number of inches, that the side lengths are all distinct, and that the side across from vertex C is 13 inches in length. How long are the other two sides? | ||

===Solution=== | ===Solution=== | ||

{{solution}} | {{solution}} |

## Revision as of 16:20, 24 November 2007

## Introductory

Are the lines , , and concurrent? If so, find the the point of concurrency.

### Solution

*This problem needs a solution. If you have a solution for it, please help us out by adding it.*

## Intermediate

*See []1992 AIME Problems/Problem 14]]*

## Olympiad

Hallie is teaching geometry to Warren. She tells him that the three medians, the three angle bisectors, and the three altitudes of a triangle each meet at a point (the centroid, incenter, and orthocenter respectively). Warren gets a little confused and draws a certain triangle ABC along with the median from vertex A, the altitude from vertex B, and the angle bisector from vertex C. Hallie is surprised to see that the three segments meet at a point anyway! She notices that all three sides measure an integer number of inches, that the side lengths are all distinct, and that the side across from vertex C is 13 inches in length. How long are the other two sides?

### Solution

*This problem needs a solution. If you have a solution for it, please help us out by adding it.*