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7.6 esercizio

Given A and C show that exist a point H such that A*H*C

Step 1

Take A,B,C such in figure and the line f and g.

Step 2

Take D and E such that A*B*D and A*C*E. Take h line through D,C and i line trough B,E. They meet in the point F for B4, because taken the triangle ABE the line h cut the side AE in the point C, so it has to cut another side and it cannot be AB because f and h already meet in D. So it exist F on BE segment.

Step 3

Take G such that G*A*B and the line k trough F and G. k meet BC because if we look at the triangle DBC it cuts DC in F so it has to cut BC, (it cannot cut BD because k and f meet in G). At this point look at the triangle ABC, k cut BC so it has to cut another side, but it can only cut AC because AB in on f and f and k meet in G. So we found the point H such that A*H*C given A and C.