There is no need to answer part 1 I just put it here since t

There is no need to answer part 1. I just put it here since the the property in part 1 might be used in part 2.

(1) Let AABC be a triangle. (a) Through each vertex, construct a line parallel to the side . The ntersection of the ine through A and the line . The ntersection of the line through C and the line . The intersection of the line through B and the line Show that A is the midpoint of B\'C, B is the midpoint of opposite the vertex. through B we denote C through A we denote B\'; through C we denote A\'. C\'A\' B\'. and C is the midpoint of A B\' C\" (b) Show that the three lines of altit ude (the lines through the vertices that are perpendicular to the opposite sides) all intersect in one point. Hint: Use the previous part and the fact from the sample midterm eram that the perpendicular bisectors of a triangle\'s sides meet in one point

Solution

let the line of altitude passing through A be called La, line of altitude passing through B be called Lb & line of altitude passing through C be called Lc.

Take the diagram given in question\'s part a for reference( just draw La, Lb & Lc for ABC).

You can see that for La:

a) La is perpendicular to BC and since BC is parallel to C\'B\' , La is perpendicular to C\'B\'.

b)La passes through A and A is the midpoint of B\'C\' from part1.

therefore, La passes through midpoint of C\'B\' and is perpendicular to it, Hence it is the perpendicular bisector of C\'B\'.
Similar conclusions can be derived for Lb & Lc.
Therefore La, Lb & Lc are the perpendicular bisectors of the sides of triangle A\'B\'C\'.
Using the property that perpendicular bisectors of a triangle meet at a point , we can conclude that La , Lb & Lc all intersect at a point.