The Ladder Theorem

 

Figure 1

I hadn't heard of the Ladder Theorem, shown in Figure 1, until I was confronted with the problem shown in Figure 2. It appeared in the thumbnail of a YouTube video and the problem was to find the value of \(x\).

Figure 2: screenshot from YouTube video

The problem looks deceptively easy but, unless you're familiar with the theorem, it's not. Using the theorem, the solution can be reached easily. Let's refer to Figure 1 (and ignore the \(x\) in Figure 2) and so we can see that \(x=4, y=10 \text{ and } z=5\). Thus \(x+y=14 \text{ and } y+z=15\). This leads to (and now the \(x\) refers to the \(x\) in Figure 2):$$
\begin{align} \frac{1}{x+19}+\frac{1}{10}&=\frac{1}{14}+\frac{1}{15}\\
\frac{1}{x+19}&=\frac{1}{14}+\frac{1}{15}-\frac{1}{10}\\&=\frac{15+14-21}{210}\\
&=\frac{8}{210}\\&=\frac{4}{105}\\x+19&=26.25\\x&=7.25 \end{align}$$A rather lengthy proof of the Ladder Theorem can by found by watching this YouTube video. It would seem that this theorem is part of a family of related theorems including:

• Ceva's Theorem
• Menelaus's Theorem
• Van Aubel's Theorem
• Steward's Theorem

I won't go into these other theorems in this post but this video discusses all of them.