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This page presents an animation that illustrates a unification of the theorems of Ceva and Menelaus. This unification is a consequence of the main result of the paper
A generalization of Routh's triangle theorem which is a joint work with Árpád Bényi.
In this paper we prove a generalization of the well known Routh's triangle theorem. Before presenting the animation we describe its main components.
The animation below is hosted in a triangle $ABC$. In the introduction to the paper we introduced a particular way of identifying cevians in $ABC$ using real numbers. We use that notation below.
There are six cevians in the animation:
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The cevian $AA_x$ is dark blue. It is fixed with $x = 0.35$.
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The cevian $BB_y$ is dark blue. It is fixed with $y = 0.3$.
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The cevian $CC_z$ is dark blue. The value $z$ belongs to $\left(\tfrac{1}{xy},+\infty\right) \cup \left(-\infty,-\tfrac{1}{xy}\right)$.
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The cevian $AA_u$ is yellow and $0 \leq u \leq x$.
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The cevian $BB_v$ is yellow and $0 \leq v \leq y$.
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The cevian $CC_w$ is yellow and $0 \leq w \leq z$.
There are three red points in the animation:
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The red point $P$ is the intersection of dark blue $AA_x$ and yellow $BB_v$.
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The red point $Q$ is the intersection of dark blue $BB_y$ and yellow $CC_w$.
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The red point $R$ is the intersection of dark blue $CC_z$ and yellow $AA_u$.
Next we describe the animation which unifies theorems of Ceva and Menelaus; the animation starts with Ceva's theorem and ends with Menelaus's theorem:
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At the beginning of the animation we have the following values: $u = x$, that is $A_u = A_x$, $v = y$, that is $B_v = B_y$, $w = z = \tfrac{1}{xy}$, $C_w = C_z$. That is the yellow and the blue cevians coincide. Since $z = \dfrac{1}{xy}$ we have $xyz = 1$. By Ceva's theorem the cevians $AA_x = AA_v, BB_y = BB_v, CC_z = CC_w$ are concurrent; they intersect at the red point. In this case the points $P, Q$ and $R$ coincide.
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As the animation proceeds $u, v, w$ decrease to $0$ with the following values at $0$: $A_0 = B$, $B_0 = C$, $C_0 = A$. The value of $z$ increases towards $+\infty$ while the point $C_z$ proceeds towards $B$. As $C_z$ passes through $B$, $z$ "switches" to $-\infty$ and it continues towards $-\tfrac{1}{xy}$.
The values of $u$, $v$, $w$, $x$, $y$, $z$ satisfy
\[
1-x y w - x v z - u y z + x y z + xyzuvw = 0.
\]
By the Corollary in our paper this equality is equivalent to $P$, $Q$ and $R$ being collinear.
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At the end of the animation
$u = v = w =0$, that is the yellow cevians $AA_0$. $BB_0$ and $CC_0$ coincide with the sides $AB$, $BC$ and $CA$, respectively. Therefore the red points $P = AA_x \cap BB_0$, $Q = BB_y\cap CC_0$ and $R = CC_z\cap AA_0$ belong to the sides $BC$, $CA$ and $AB$, respectively. At the end of the animation we have $z = -\tfrac{1}{xy}$, that is $xyz=-1$. By Menelaus's theorem $P, Q$ and $R$ are collinear in this case.
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