Question

Given: BE and AD are altitudes (intersecting at H) of triangle ABC, while F,G, and K are midpoints of AH, AB and BC, respectively. Prove: angle FGK is a right angle.

Answer 1

Since
`BE` is an altitude of a triangle
`ABC`, it is perpendicular to
`AC` (since the triangle ).

Since
`G` and
`K` are the midpoints, then the line
`BG=GA` and the line
`BK=KC`. The line
`KG` must be parallel to the line
`AC` and is therefore perpendicular to
`BE.`

Now since F and
`G` are the mid points, we get
`GA=BG`. The line
`GF` is parallel to BH and is perpendicular to AC.

Therefore, we have

`KG-GK` parallel to
`AC`

`AC` perpendicular to
`BE-BH-BJ`

`BE-BH-BJ` parallel to
`GF`

Thus
`GK` is perpendicular to
`GF` and also ∠
`FGK` is a right angle.