Triangles, Quadrilaterals & Polygons

Area $=\dfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$ $=\dfrac12\left|\begin{smallmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{smallmatrix}\right|$. Collinear iff this is $0$. The same determinant gives the condition three points lie on a line.

• Vertices on a curve: for points $\big(x_i,\tfrac1{x_i}\big)$ on $xy=1$ the determinant factorises to $\dfrac{|(x_1-x_2)(x_2-x_3)(x_3-x_1)|}{2\,x_1x_2x_3}$; for $(x_i,x_i^2)$ on a parabola it is $\tfrac12|(x_1-x_2)(x_2-x_3)(x_3-x_1)|$ — Vandermonde shapes.
• Mensuration the coordinate questions borrow: an equilateral triangle of side $a$ has area $\tfrac{\sqrt3}{4}a^2$ and altitude $\tfrac{\sqrt3}{2}a$; the segment joining the midpoints of two sides is parallel to the third side and half its length (midpoint theorem), so the midpoint triangle has a quarter of the area.

First, what $aA+bB+cC$ means: do it one coordinate at a time. The x-coordinate is $ax_1+bx_2+cx_3$, the y-coordinate is $ay_1+by_2+cy_3$. Adding points, and multiplying a point by a number, always works coordinate-wise in this chapter.

• Centroid: $\left(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right)$ — the plain, unweighted average.
• Incentre: $\dfrac{a\,A+b\,B+c\,C}{a+b+c}$, where $a,b,c$ are the side lengths opposite $A,B,C$. You must compute those three lengths first; they are the weights.
• Circumcentre: equidistant from all three vertices — solve two perpendicular-bisector equations. For a right triangle it is simply the midpoint of the hypotenuse.

• Recover a vertex from a midpoint: if $M$ is the midpoint of $BC$, then $B+C=2M$, so $C=2M-B$. The factor of $2$ is the whole point — $M$ is the average of $B$ and $C$.
• A median joins a vertex to the midpoint of the opposite side. So being given a median is being given a midpoint, and a midpoint hands you the missing vertex by the line above.
• An altitude from a vertex is perpendicular to the opposite side: its slope is the negative reciprocal of that side's slope, and it passes through the vertex.
• Special triangles: an equilateral or isosceles condition fixes the third vertex, usually through equal distances. The third vertex of an equilateral triangle on a given base generally has irrational coordinates.
• Isosceles triangle from its apex angle: if the apex angle is $A$, each base angle is $90°-\tfrac A2$, so $\tan(\text{base angle})=\cot\tfrac A2$; conversely with $t=\tan(\text{base angle})$, $\tan A=\tan(180°-2\cdot\text{base})=-\dfrac{2t}{1-t^2}$. When the sides through the apex are given as lines, their slopes give $\tan A$ and this recovers the base direction.

• Parallelogram $ABCD$: the diagonals join opposite vertices, $AC$ and $BD$, and they bisect each other. So $A+C=B+D$, giving $D=A+C-B$, and they meet at the midpoint of either.
• Side vector: the vector from $A$ to $B$ is just $B-A$, computed coordinate by coordinate. Write $\vec{AB}=(u_1,v_1)$ and $\vec{AD}=(u_2,v_2)$ for the two sides meeting at $A$.
• Area $=|u_1v_2-u_2v_1|$. These are components of the two side vectors — note that everywhere else in this chapter $(x_1,y_1)$ means a vertex, so do not feed vertices into this formula.
• A square or rectangle built on two given parallel sides uses the perpendicular distance between them as the other side length. A square with diagonal $d$ has area $\tfrac{d^2}{2}$ and side $\tfrac{d}{\sqrt2}$.
• Telling the quadrilaterals apart by their diagonals (compute both diagonal vectors first): a parallelogram whose diagonals are perpendicular ($\vec{AC}\cdot\vec{BD}=0$) is a rhombus; one whose diagonals are equal in length is a rectangle; both together make a square. A non-parallelogram with equal diagonals and one pair of parallel sides is an isosceles trapezium. "Rhombus, not rectangle" is the key when the diagonals are perpendicular but unequal.