MHB Why does this proportion work?

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The discussion explains the relationship between the dimensions of triangle XYZ inscribed in circles and the similarity of triangles OYX and OXZ. By Thales' theorem, the angles formed indicate that both triangles are right triangles, leading to the conclusion that they are similar. The proportion OY/OX = OX/OZ is derived from the similarity of these triangles, allowing for the equation (x-9)/(x-5) = (x-5)/x. This mathematical relationship demonstrates how the properties of similar triangles can be applied to understand the proportions in the diagram. The explanation clarifies the geometric reasoning behind the proportion's validity.
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In this diagram, triangleXYZ is inscribed into the circles. O is the center of the larger circle. OZ=x, altitude XO=x-5, and OY=x-9. angleXOZ and angleXOY are both right angles. Using the two similar right triangles OYX, and OXZ, this proportion can be written: OY/OX=OX/OZ
Then: (x-9)/(x-5)= (x-5)/x

My daughter wants to know why this works. How was this proportion written, why it works and how we know triangleOYX is similar to triangleOXZ? We appreciate any information as to why this works.

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By Thales' theorem, $\angle{YXZ}=90^\circ$. As $\angle{XYO}+\angle{XZO}=90^\circ$ and $\angle{XYO}+\angle{YXO}=90^\circ$, $\angle{XZO}=\angle{YXO}$ and triangles $OYX$ and $OXZ$ are similar.
 
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