11: Similar Triangles

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To understand the remaining theorems of Circle we need to know the property of SIMILAR TRIANGLES. This is very important and very useful for solving various types of problems. In similar triangles, the angles are equal i.e. both the triangles have the same set of three angles. 

And the theorem states that the sides opposite to the equal angles are in the same ratio. 

Now, if you look at the diagram, the ratios shown are equal since they are framed as per this theorem. 


Now we look at two triangles which are similar but look somewhat different from the above diagram
The diagram shows the three sides of the two triangles who are opposite to equal angles. 

​Thus, the ratios are equal as shown. 


Now we look at two triangles which do not look similar but are similar.
​Please pay attention

The triangles ADC and BDC are similar because angles BAC and DBC are shown to be the same while angle C is common to both the triangle. And if 2 angles are equal the 3rd angle has to be equal because the 3 angles always add up to 180 degrees. The rest of the explanations shown in the diagram are self explanatory.

TWO THEOREMS OF CIRCLE
​the proofs of which are based on SIMILAR TRIANGLES

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This theorem proves that when two chords AD and BC meet at E, the product of the segments of the chords are the same. This is very interesting and very useful in solving problems. 

The proof depends on Similar triangle theorem as shown in the diagram. Please pay attention, draw it out yourself and understand thoroughly. 


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Finally the last theorem of Circle. 

Please follow along the instructions in the diagram yourself and satisfy yourself about the result. The proof comes from the similar triangle theorem as shown above while the result is quite simple and needs to be committed to memory and used subsequently in problem solving.