• $(x_1 y_1)$

• $(x_2 y_2)$

• $(x_3 y_3)$

$(\frac{y - \frac{(y_1 + y_2)}{2}}{x-\frac{(x_1 + x_2)}{2}}) \times \frac{(y_2 - y_1)}{(x_2 - x_1)}$

$=-1$

$r=\sqrt{(x_0 - x_2)^2 + (y_0 - y_2)^2}$

B1: $(y_0 - \frac{(y_1 + y_2)}{2}) (y_2 - y_1) + (x_0 - \frac{(x_1 + x_2)}{2}) \times (x_2 - x_1) = 0$

B2: $(y_0 - \frac{(y_2 + y_3)}{2}) (y_3 - y_2) + (x_0 - \frac{(x_2 + x_3)}{2}) \times (x_3 - x_2)=0$

$(x_0, y_0)$ is the centre of the circle which passes through $p_1$, $p_2$ and $p_3$.

Show that the four points (1, 0), (2, -7), (8, 1) and (9, -6) are con cyclic.

Solution: $(\frac{y + 3.5}{x - 1.5}) \times -7 = -1$

$y + 3.5 = \frac{x}{7} - \frac{1.5}{7}$

$y - 0.5 = -7x + 30.5$

$(\frac{y - 0.5}{x - 4.5}) \times \frac{1}{7} = -1$

$y_0 + 3.5 = \frac{x_0}{7} - \frac{1.5}{7}$

$y_0 - 0.5 = -7 x_0 + 31.5$

$\frac{x_0}{7} - \frac{1.5}{7} - 3.5 = -7 x_0 + 32$

$\frac{50 x_0}{7} = \frac{224 + 1.5 + 24.5}{7}$

$=\frac{250}{7}$

$x_0 = 5$

$\therefore y_0 =-7 x_0 + 32$

$=-7 \times 5 + 32$

$=-3$

$\therefore$ the centre of the circle is at (5, -3).

radius $r = OP_1 =\sqrt{(5-1)^2+(-3)^2} =5$

$(x-x_0)^2 + (y-y_0)^2 = r^2$

i.e., $(x-5)^2+(y+3)^2=25$

Check to see if (9, -6) lies on this circle.

$(9-5)^2 + (-6+3)^2 = 16 + 9 = 25$

$P_1(x_1, y_1), P_2(x_2, y_2)$ and $P_3(x_3, y_3)$

GENERAL FORM $C: x^2 + y^2 + 2gx + 2fy + c = 0$

since $P_1$ lies on C $\Rightarrow x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = 0$

$x_2^2 + y_2^2 + 2gx_2 + 2fy_2 + c = 0$

$x_3^2 + y_3^2 + 2gx_3 + 2fy_3 + c = 0$

$\frac{(y_2-y)}{(x_2-x)} \times \frac{(y_1-y)}{(x_1-x)}$

$=-1$

$(x-x_1) (x-x_2) + (y-y_1)(y-y_2)= 0$

$x^2 + y^2 - (x_1 + x_2) x - (y_1 + y_2) y + (x_1 x_2 + y_1 y_2) = 0$

$x^2 + y^2 + 2gx + 2fy + c = 0$

P(a, b)

$C:x^2 + y^2 + 2gx + 2fy + c = 0$

$r = \sqrt{g^2 + f^2 - c}$

P is inside the circle if and only if

$\sqrt{(a+g)^2 + (b+f)^2}<\sqrt{g^2 + f^2 - c}$

$a^2 + b^2 + 2ag + 2bf + c < 0$

Given a point P(a, b) and a circle,

$C:x^2 + y^2 + 2gx + 2fy + c = 0$

$a^2 + b^2 + 2ga + 2fb + c$

$a^2 + b^2 + 2ga + 2fb + c = 0 \Rightarrow$ P lies on the circle C

$a^2 + b^2 + 2ga + 2fb + c < 0 \Rightarrow$ P lies inside C

$a^2 + b^2 + 2ga + 2fb + c > 0 \Rightarrow$ P lies outside C

Thank you