How to calculate the two tangent points to a circle with radius R from two lines given by three points

Question

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$.

Sketch would explain the problem more. I need to find the tangent points $A(x_a,y_a)$ and $B(x_b,y_b)$. Note that the center of the circle is not given. Please help.

Answer 1

Let $O$ be the centre of the circle. Consider $\triangle OAP$. Find $\angle APO$ using the cosine rule and the fact that $\angle QPR=2\angle APO$. We know that $\angle OAP=90^\circ$ and $|OA|=r$. Solve.

Answer 2

Since you need the coordinates of the two points I'll give a more analytic solution. Choose your coordinates so that $PR$ lies on the $x$-axis. Introduce two unknowns, for example $x_b$ and $x_a$. Write the equation of the line through $P$ and $Q$. Express the coordinates of $A$ and $B$ in terms of the unknowns. It is easy to write the equations of the lines through $A$ and $B$, perpendicular to $PQ$ and $PR$, respectively. Their intersection is $O$, the centre of the circle, which must satisfy the equation of the angle bisector of $\angle QPR$. The last condition you need is that distance $OB$ is $r$. Solve the system.

Answer 3

My inclination would be to write the equations of the two lines, then use the point-to-line distance formula (the distance between $Ax+By+C=0$ and $(x,y)$ is $\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}$) to write a system of two equations (the distance from the desired point to each line is $r$) in two unknowns (the coordinates of the desired point). The system will have four solutions because it's based on the whole lines, not just the rays you've got in your picture.

Answer 4

An identity that might prove useful in this problem is $$ \cot\left(\frac{\theta}{2}\right)=\frac{\sin(\theta)}{1-\cos(\theta)}=\frac{1+\cos(\theta)}{\sin(\theta)}\tag{1} $$ In $\mathbb{R}^3$, one usually uses the cross product to compute the sine of the angle between two vectors. However, one can use a two-dimensional analog of the cross product to do the same thing in $\mathbb{R}^2$.

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In the diagram above, $(x,y)\perp(y,-x)$ and so the $\color{#FF0000}{\text{red angle}}$ is complementary to the $\color{#00A000}{\text{green angle}}$. Thus, $$ \begin{align} \sin(\color{#FF0000}{\text{red angle}}) &=\cos(\color{#00A000}{\text{green angle}})\\[6pt] &=\frac{(u,v)\cdot(y,-x)}{|(u,v)||(y,-x)|}\\[6pt] &=\frac{uy-vx}{|(u,v)||(x,y)|}\tag{2} \end{align} $$ $uy-vx$ is the normal component of $(u,v,0)\times(x,y,0)$; thus, it is a two dimensional analog of the cross product, and we will denote it as $(u,v)\times(x,y)=uy-vx$.

Let $u_a=\frac{Q-P}{|Q-P|}$ and $u_b=\frac{R-P}{|R-P|}$, then since $$ |A-P|=|B-P|=r\cot\left(\frac{\theta}{2}\right) $$ we get $$ \begin{align} A &=P+ru_a\cot\left(\frac{\theta}{2}\right)\\ &=P+ru_a\frac{1+u_a\cdot u_b}{|u_a\times u_b|} \end{align} $$ and $$ \begin{align} B &=P+ru_b\cot\left(\frac{\theta}{2}\right)\\ &=P+ru_b\frac{1+u_a\cdot u_b}{|u_a\times u_b|} \end{align} $$ where we take the absolute value of $u_a\times u_b$ so that the circle is in the direction of $u_a$ and $u_b$.