Taking it to the next level
In triangle, find the sum of the total number of triangles(excluding degenerate triangles) and quadrilaterals(excluding degenerate quadrilaterals) in the given figure.
Note : and are perpendicular to .
Line segment FH is parallel to AB.
G is a point on the other side of FILH compared to AB
There are 30 additional points on the line segment IL, which are connected to both A and B with line segments.
There are 27 additional points on the line segment FG, which are connected to H with line segments
There are 27 additional points on the line segment GH, which are connected to F with the line segments.
Line segment EC is parallel to AB.
D is a point on the other side of EJKC compared to AB
There are 30 additional points on the line segment EC, which are connected to both A and B with line segments.
There are 27 additional points on the line segment ED, which are connected to C with line segments
There are 27 additional points on the line segment CD, which are connected to E with line segments.
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1 solution
The total number of t r i a n g l e s = [ ( 1 2 7 8 5 ∗ 2 ) + 6 ] + [ 2 ∗ 1 1 2 8 6 ] = 2 5 5 7 6 + 2 2 5 7 2 = 4 8 1 4 8 .
The total number of q u a d r i l a t e r a l s = 3 0 + [ ( 2 9 + 2 8 + 2 7 + 2 6 + 2 5 + 2 4 + 2 3 + 2 2 + 2 1 + 2 0 + 1 9 + 1 8 + 1 7 + 1 6 + 1 5 + 1 4 + 1 3 + 1 2 + 1 1 + 1 0 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 ) − 1 ] + 3 + [ 2 9 + 2 8 + 2 7 + 2 6 + 2 5 + 2 4 + 2 3 + 2 2 + 2 1 + 2 0 + 1 9 + 1 8 + 1 7 + 1 6 + 1 5 + 1 4 + 1 3 + 1 2 + 1 1 + 1 0 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 ] + 1 + [ 2 ∗ 4 2 9 4 9 6 7 2 9 4 0 ] + 1 1 0 + [ 2 ∗ 1 3 1 0 4 4 ]
= 3 0 + 4 3 4 + 3 + 4 3 6 + 8 5 8 9 9 3 4 5 8 8 0 + 1 6 3 4 5 6 2 1 9 + 1 1 0 + 2 6 2 0 8 8
= 8 6 0 6 3 0 6 5 2 0 0
Therefore, the total number of t r i a n g l e s and q u a d r i l a t e r a l s
= 8 6 0 6 3 0 6 5 2 0 0 + 4 8 1 4 8
= 8 6 0 6 3 1 1 3 3 4 8