Law of Cosines: Proof Without Words

geometry trigonometry

Solution 1:

The image was a little difficult for me to parse at first, so here's a refinement:

enter image description here

Now ... With $A$ the vertex opposite $a$ in the $a$-$b$-$c$ triangle, we can express the power of $A$ with respect to the circle as two chord-chord products to get

$$(2a\cos\theta-b)\cdot b = (a-c)\cdot(a+c)$$

and the result follows. $\square$

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