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how to find the distance of the incenter of an equlateral triangle to mid center of each side?

I am given a diagram of a equalteral triangle with all bisectors drawn, the incenter and the circumcenter are one and the same. I am also given the distance of the circum center from the vertexes. I am told to find the distance from the incenter to the midpoint of each side, how do I solve it?

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Oops, i meant 180/3 = 60.

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4 Answers

The distance of the circumcenter to the vertex is two thirds of the altitude. The length you are trying to find is the other third. Just halve the distance you already know. 

The best thing to do is to draw this triangle on your paper first. If you draw each of the angle bisectors to the incenter, you will create 6 right triangles. Since the angle bisectors create a 30 degree angle with the side of the triangle, and the angle bisector creates a 90 degree angle where it bisects the side of the triangle, you have a 30-60-90 triangle. If the distance from the incenter to the vertex is given... lets let that = c. This is the hypontenuse of the right triangles you created in your drawing. The distance to the midpoint is the smallest side of your right triangle, so this would be .5c using the properties of a 30-60-90 triangle.

If the triangle is equilateral, then you form a 30, 60, 90 triangle. Use the bisector to the vertex as a hypotenuse with length equal to two times the length of the shortest side. The shortest sideis the distance from the circumcenter to the the vertexes. All you have to do is take the distance given to you and divide it by two.

The first thing you have to keep in mind is the symmetry of the problem. Here you have an equiliateral triangle, meaning all sides have the same length and all angles measure the same. That gives you the first clue, which is that the angles of the triangle are all 60 degrees (360/3).

The second thing you have to realize is that any line from a vertex to the circumcenter will bisect the angle of 60 degrees into two 30 degree angles. Put this together with the fact that a line that goes through the midpoint of one of the sides of the triangle and the circumcenter.

If you draw all this out you will find out that you wind up with a nice set up to the problem, where you are looking for the length of one of the sides of a right triangle.

The final step is to use trigonometry. You'll have a right triangle with an angle of 30 degrees, and a hypotenuse equal in length to the distance between the circumcenter and the vertex. Solve for the side they are asking about (think about the sine and cosine definitions), and you should be set.