How many different ways to connect two triangles exist

combinatorics

Let's say you have two triangles with of side's lengths $a, b, c$ and $x, y, z$ respectively, such that all lengths are different.

Two triangles are connected if they have one common vertex and two sides lie on one straight line.

Question

How many different ways to connect two triangles exist?

Edit. My answer is 54.

Let $A$, $B$, $C$ and $A_1$, $B_1$, $C_1$ be the vertex name of triangles $\Delta ABC$ and $\Delta A_1B_1C_1$ respectively.

One can fix the first triangle $\Delta ABC$ and connect vertices $A$ and $A_1$ such that sides $AC$ and $A_1C_1$ will lie on one straight line.
One put the second triangle in three ways: a) outside, b) inside, c) near:

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In three cases above one can rotate $3$ times the second triangle $\Delta A_1B_1C_1$ while the vertex $A$ is fixed. The triangle $\Delta ABC$ has the two vertices on each side, therefore, one can connect two triangles on one side by $3 \times 2$ ways. The fixed triangle has three sides, therefore, one can connect two triangles by $3 \times 2 \times 3 =18$ ways.

One can have three cases above, finally, $18 \times 3 =54$ ways.

Edit 2. After the @SheridanGrant's answer I added the case (d).
And I thinking about the case (f). Should the case (f) give additional ways to the solution? Triangles have the same height.

enter image description here

Best Answer

You're missing a case--in case $c$, reflect the blue triangle over its bottom side. 4 cases, two edges at each side, 3 choices of vertex for first triangle, 3 choice of vertex for second triangle yield $4 \cdot 2 \cdot 3 \cdot 3 = 72 = 54 \cdot \frac{4}{3}$.

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