Chapter 4: Geometry

FIGURE 4.12
Left: Ceva’s theorem. Right: Menelaus’s theorem.
The hypotenuse is a diameter of the circumscribed circle. The median joining
the midpoint of the hypotenuse (the center of the circumscribed circle) to the right
angle makes angles ¾ and ¾ with the hypotenuse.
Additional facts about triangles:
1. In any triangle, the longest side is opposite the largest angle, and the shortest
side is opposite the smallest angle. This follows from the law of sines.
2. Ceva’s theorem (see Figure 4.12, left): In a triangle , let , and
be points on the lines , and , respectively. Then the lines ,
and are concurrent if, and only if, the signed distances , ,...
satisfy
¡ ¡ ¡ ¡ (4.5.5)
This is so in three important particular cases: when the three lines are the
medians, when they are the bisectors, and when they are the altitudes.
3. Menelaus’s theorem (see Figure 4.12, right): In a triangle , let , and
be points on the lines , and , respectively. Then , and
are collinear if, and only if, the signed distances , , . . . satisfy
¡ ¡ ¡ ¡ (4.5.6)
4. Each side of a triangle is less than the sum of the other two. For any three
lengths such that each is less than the sum of the other two, there is a triangle
with these side lengths.
5. Determining if a point is inside a triangle
Given a triangle’s vertices È ¼ , È ½ , È ¾ and the test point È ¿ , place È ¼ at the
origin by subtracting its coordinates from each of the others. Then compute
(here È ´Ü Ý µ)
Ü ½ Ý ¾ Ü ¾ Ý ½
Ü ½ Ý ¿ Ü ¿ Ý ½
(4.5.7)
Ü ¾ Ý ¿ Ü ¿ Ý ¾
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