Let be an triangle. On side of the triangle let be two points so that . Let and . Let be the midpoints of the segments: . Proof that can be the three sides of a triangle. Find a condition so that .

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- Jan 9th 2013, 05:38 AMmathmagixtriangles, midpoints and parallelism
Let be an triangle. On side of the triangle let be two points so that . Let and . Let be the midpoints of the segments: . Proof that can be the three sides of a triangle. Find a condition so that .

- Jan 11th 2013, 10:36 AMkalyanramRe: triangles, midpoints and parallelism
Attachment 26533

With out any loss of generality we consider an oblique co-ordinate system with the origin as A with B and C along x and y axis such that BC=a, AC=b, AB=c the lengths of the given triangle. Now say AD = AB, BE = AB where are some constants. Using the condition we have .

Using Basic Proportionality Theorem in this triangle and the formula for internal division of a line segment according to a given ratio. We can determine the co-ordinates of the points as

.

Use the Cosine Rule to get the inclination between the co-ordinate axes say as .Now use the distance formula in the oblique system to determine the lengths and show that the sum of lengths of any two sides is greater than the third side so they can form a triangle. will prove to be helpful.

Use the slope equation to get the condition between for .

Kalyan - Jan 11th 2013, 08:11 PMjohngRe: triangles, midpoints and parallelism
This partial solution is similar to the above, but the algebra for the first question was beyond me. As implied above, MP is parallel to AB iff length AD = length EB. I put the given figure in a dynamic geometry program which was very helpful. Here's the image.

Attachment 26540 - Jan 13th 2013, 10:54 PMLoidaWardRe: triangles, midpoints and parallelism
As implied above, MP is parallel to AB iff length AD = length EB.

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