Construction of Parallel Line

Given line AB and point P, not on AB. Construct a line through P, parallel to AB. 

1. Draw a line through P, so that it intersects line AB at point D. 
2. Open the compass to a comfortable distance, set the compass at D, and make an arc so that it intersects DP at E and line AB at F. 
3. Using the same compass opening, make an identical arc with the compass set at P, so that the arc intersects DP at G. 
4. Set the compass at E, and make an arc so that it intersects line AB at point F. 
5. Using the same compass opening, make an identical arc with the compass set at G so that it intersects the arc you made in step 3 at point H. 
6. Draw a line through P and H , line PH is parallel to AB.

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Appolonius Theorem

In a triangle , the sum of the square of two sides of a triangle is equal to twice the sum of the square of the median which bisects the third side and the square of half the third side. In triangle ABC , if AD is a median then AB² + AC² = 2(AD²+DC²) or 2(AD²+BD²)

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