Proof circle theorem: when two chord intersect, the product of parts of the chord are equal.
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Today we'll prove the circle chord intersection theorem. When two chords intersect inside a circle, the products of their segments are equal. Here we have chords AB and CD intersecting at point P, creating segments AP, PB, CP, and PD. The theorem states that AP times PB equals CP times PD.
To prove our theorem, we focus on triangles APC and DPB formed by the intersecting chords. Triangle APC is highlighted in blue, and triangle DPB in red. Notice that these triangles share vertical angles at point P, which are equal. Our strategy is to prove these triangles are similar, which will give us the proportional relationships we need.
Now we use the inscribed angle theorem. Angles CAD and CBD are inscribed angles that subtend the same arc CD, shown in blue. By the inscribed angle theorem, these angles are equal. Similarly, angles ACD and ABD subtend the same arc AD, shown in red, making them equal as well. These angle relationships are crucial for proving triangle similarity.
Now we prove triangle similarity using the AA criterion. We have established that angle APC equals angle DPB because they are vertical angles. We also showed that angle CAP equals angle DBP using inscribed angle relationships. With two pairs of equal angles, triangles APC and DPB are similar by the AA similarity criterion. This similarity gives us proportional corresponding sides.
From the similarity of triangles APC and DPB, we know that corresponding sides are proportional. This gives us the ratio AP over DP equals CP over BP. Cross multiplying this proportion, we get AP times BP equals CP times DP, which is the same as AP times PB equals CP times PD. This completes our proof of the circle chord intersection theorem.