PointXbelongs to the interior of a triangleABC. The distance from this point to linesAB, BC, ACisk, l, mrespectively andris the length of an inradius of this triangle. Prove that:

Thanks in advance!

Printable View

- January 29th 2011, 09:44 AMterrogeometric proof - triangle
Point

*X*belongs to the interior of a triangle*ABC*. The distance from this point to lines*AB, BC, AC*is*k, l, m*respectively and*r*is the length of an inradius of this triangle. Prove that:

Thanks in advance! - February 2nd 2011, 08:38 PMabhishekkgp
- February 7th 2011, 05:03 PMJoshuaJava
Okay maybe I'm crazy but does this prove it? :

If point X is at the centre of the incircle then the distances from this point to each line are the shortest possible and are also equal to r. Therefore, k + l + m = 3r ..... therefore k + l + m > 2r

Somebody tell me if this is right please, and if it is not, please tell me why. Thanks. - February 7th 2011, 07:08 PMabhishekkgp
- February 8th 2011, 01:09 AMFlatiron
- February 8th 2011, 08:35 AMJoshuaJava
Because k l and m are at the centre of the incircle.

- February 8th 2011, 08:38 AMJoshuaJava
It does cover the general case because if the point is at the centre of the incircle, it means that if it were anywhere else, the sum of each length would be greater than the sum of each length from the cente, therefore it would definately be greater than 2r at any other point in the triangle.

- February 8th 2011, 08:41 AMabhishekkgp
- February 8th 2011, 09:54 AMArchie Meade
At the incentre, the triangle area can be expressed as the sum of the areas of 3 internal triangles

At the point, the triangle area can be expressed as the sum of the areas of 3 other internal triangles

Then

Since the length of any side of a triangle is less than the sum of the other two sides, then

Therefore the inequality is true - February 9th 2011, 07:34 AMJoshuaJava
Damn... I don't know. But I'm sure there is a way to prove it. Sorry for wasting your time though.

- February 9th 2011, 10:54 AMArchie Meade
- February 13th 2011, 08:36 AMJoshuaJava
Yes, but, as I said, if it is at the incentre, the sum of the lengths is smaller than anywhere else in the triangle, but I don't know how to prove this, but it is true.

- February 13th 2011, 08:54 AMArchie Meade
The incentre is one of infinitely many points in the triangle.

There is no reference to it in the original post at the start of the thread.