In triangle ABC Draw the median AM. K is a point on AM such that: AK/AM=1/3, BK intersects AC at N. a. Calculate the area of ​​triangle AKN, knowing the area of ​​triangle ABC is S. b. A

Question

In triangle ABC Draw the median AM. K is a point on AM such that: AK/AM=1/3, BK intersects AC at N.

a. Calculate the area of ​​triangle AKN, knowing the area of ​​triangle ABC is S.

b. A line through K intersects the sides AB and AC at I and J respectively. Prove that AB/AI + AC/AJ = 6

Answer 1

Here are some suggestions on part b. There might be a way to do it using theorems about medians of a triangle and similar triangles, but I do not see it. I was able to reach the conclusion using coordinate geometry (same coordinates used in video).

1. Find the x coordinate of point I (in terms of x1, y1, x2) by finding the equation of the line passing through K with a slope designated by some letter (I used q). Find the y coordinate by plugging that value into the equation of AB (which is y = (y1/x1)x.
2. Do the same for point J (this is quite a bit prettier than the coordinates of I, especially because the y coordinate is 0).
3. Use distance formula to find d(A,B), d(A,I). d(A,C), d(A,J).
4. Set up the expression AB/AI + AC/AJ and simplify. Amazingly everything cancels out except a 6, which is the target.
5. It would likely be possible to determine restrictions on the slope of the line through K, but I did not bother to do so.

This Desmos graph might prove useful, but there are not any notes.

desmos.com/calculator/cx8tnhqowy