Triangle centers

Japanese school year: Math A

What you learn

Study the definitions and geometric properties of the five triangle centers: circumcenter, incenter, centroid, orthocenter, and excenters. This topic requires foundational knowledge of angle bisectors, medians, and perpendiculars. These concepts are essential for advanced Euclidean geometry, structural balancing in mechanics, and analyzing geometric configurations.

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Key points

This describes the position of centroid $G$ . It divides the median line drawn from vertex $A$ to the opposite midpoint $M$ in a $2:1$ ratio.

AG : GM = 2 : 1

Known as the median theorem (Apollonius' theorem), this relates sides $AB, AC$ to median $AM$ and half-side $BM$ , where $M$ is the midpoint of $BC$ .

AB^2 + AC^2 = 2(AM^2 + BM^2)

This is the angle bisector theorem. When the bisector of angle $A$ meets opposite side $BC$ at $D$ , the divided segment ratio $BD:DC$ equals the adjacent side ratio $AB:AC$ .

\frac{BD}{DC} = \frac{AB}{AC}

This distinguishes the circumcenter and incenter. The circumcenter is the center of the circumcircle and equidistant to all vertices, while the incenter is the center of the incircle and equidistant to all sides.

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Practice 1Coming soon