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28 Oct
Posted by Rijul Saini in Uncategorized . Tagged: Geometry , Isogonal Conjugates , Olympiad , Symmedian . 1 comment
In a , if two cevians and where are drawn such that , then and are said to be Isogonal to each other, i.e. they are called Isogonal lines.
The first theorem one should study regarding Isogonal Lines is the following:
Theorem 1: Steiner’s Theorem: In a , the two cevians are isogonal if and only if
Theorem 1: Steiner’s Theorem:Proof of Steiner Theorem: I’ll prove the forward direction, the backward case is left to the interested reader. Let We have in , by the rule of sines, Similarly, by applying rule of sines in , Multiplying, we get, again, by the rule of Sines. This completes out proof. End of Proof of Steiner’s Theorem
Theorem 2: Isogonal Conjugate Theorem If in a the cevians are concurrent, and if are isogonal to respectively, then are concurrent. If concur at , and concur at , then is said to be the Isogonal Conjugate of . Therefore, one can restate the theorem as saying that every point has an Isogonal Conjugate, which is , of course, unique. Proof of Isgonal Conjugate Theorem: We have, by Steiner’s theorem,
Theorem 2: Isogonal Conjugate TheoremMultiplying, we get, But, by Ceva’s Theorem, since are concurrent. Therefore, we have Therefore, are concurrent, by Converse of Ceva’s Theorem. End of Proof of Isogonal Conjugate Theorem
Now, it’s time to derive some important Corollaries from the above. But first we define what a symmedian is. A Symmedian is the cevian which the Isogonal Conjugate of the Median. Also, the three Symmedians of a triangle concur (From theorem 2) at the Symmedian Point (Also called the Lemoine Point).
Theorem 3: Corollary to Steiner’s Theorem In a , cevian is a symmedian if and only if Proof The proof follows directly from Steiner’s Theorem, by putting the ratio equal to . End of Proof
Theorem 4: Prove that in any triangle, the orthocentre and the circumcentre of those triangles are Isogonal Conjugates. (The proof, which involves very simple angle chasing, is left to the interested reader.)
Theorem 4:Before going on to the next Theorem regarding Symmedians, we first define what we mean by the Intouch Triangle and the Gergonne Point. The Intouch Triangle is the triangle whose vertices are the points where the incircle of a triangle meets the sides of that triangle. The Gergonne Point is the point of concurrency of the lines joining a point of a triangle to the point where the incircle touches the side opposite to it. Thus, it’s the perspectrix of the main triangle and its Intouch Triangle.
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