Geometry: Power of a Point & Mass Points

Study Sheet

Geometry: Power of a Point & Mass Points

Circle power and weighted balancing

Power of a Point

Tip
The constant product

From a point PP, along any line through PP meeting a circle at A,BA,B, the product PA⋅PBPA\cdot PB is constant; for a tangent of length tt, t2=PA⋅PBt^2 = PA\cdot PB.

In plain terms. No matter which line you draw through a point, the two distances to the circle multiply to the same number.

Example. Chords ABAB and CDCD cross at PP with PA=2,PB=6,PC=3PA=2, PB=6, PC=3: then PD=2⋅63=4PD = \dfrac{2\cdot 6}{3} = 4.

Mass Points and Coordinates

Concept
Mass points

Hang weights at the vertices so each side balances at its division point (weight ×\times length equal on both sides); the weights reveal all segment ratios.

In plain terms. Treat the triangle like a mobile: put weights so it balances, and the balance conditions give the ratios you want.

Example. If a cevian splits a side 2:32:3, put weights 33 and 22 at its endpoints so 3⋅2=2⋅33\cdot 2 = 2\cdot 3 balances at the division point.

Concept
When in doubt, use coordinates

Place the figure on axes; use the distance formula, perpendicular slopes, and the shoelace area formula.

In plain terms. Give the points coordinates and the geometry turns into algebra you can just compute.

Example. A right angle at the origin with legs on the axes makes the third vertex, lengths, and area immediate.

Reminder — Distance and midpoint:  d=(x2−x1)2+(y2−y1)2,M=(x1+x22, y1+y22)d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}, \qquad M=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)