Radical Axis to the Two Circles

The radical axis to the two circles is a moving point that enables the tangent segments drawn from it to the two circles to be of equal length.

The equation for the radical axis of two circles is:

S − S’ = 0

Where S ≣ x2 + y2 + 2gx + 2fy + c = 0 and S’ ≣ x2 + y2 + 2g’x + 2f’y + c’ = 0

It is the equation for the common chord of two circles if S and S’ intersect at two real and distinct points.

If S’ = 0 and S = 0 touch, S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.

Set the x2 and y2 coefficients to unity in the equation of the two circles to obtain the radical axis.

What is the importance of the radical axis of the two circles?

The radical axis has equal tangents to two circles. Hence, there is a special point along the radical axis where the two equal tangents create a straight line. The unique point of the radical axis is the midpoint of a tangent to both circles. In addition to the homothetic centres along the centre line, each of these two sites is a specific homothetic centre. The routine method may produce a common external tangent between the two circles.

Properties

• Three circles’ radical axes are concurrent.

• The radical axis and common chord are similar.

• If two circles intersect the third circle orthogonally, the two circles’ radical axis cuts the third circle.

• The concurrent radical axis of the three circles is the radical centre.

• The circles that do not have a radical axis are all concurrent.

• A pair of circles with the same radical axis is a coaxial system.

What is the symmetry of a circle?

A circle’s diameter is a line across its centre, and a circle’s diameter is thus the symmetry line that splits it in half. The number of symmetry lines in a circle is infinite.

Lines of symmetry for circles

A circle has infinite symmetry. Compare this to polygons like triangles and quadrilaterals Symmetry Lines. The circle is the most symmetrical two-dimensional figure, so it is familiar. Coins, clock faces, wheels, and the full moon in the sky are all examples of circles.

Limiting point

An inversion point is a point where two circles generate concentric circles. A coaxial system’s limiting points are inverse points with respect to any of the system’s circles, and every circular pair has two limiting points.

Radical axis construction

A line between any two points on two circles, A and B, build the radical axis. Draw a circle C that intersects both A and B twice. A and C and B and C are the two lines that intersect. Point J is the radical centre of all three circles. As a result, this point is also on the radical axis of A and B, and repeating the procedure with a second similar circle D results in a second point K. It’s the path that passes through J and K.

Lines intersecting anti-homologous points create the radical axis of two circles. P and Q, as well as S and T, are anti-homologous. These are the four points on a circle that intersects both circles.

A special case to this approach is anti-homologous points from either an internal or external centre of similarity. Consider two rays emerging from a homothetic centre E on the external side. P and Q and S and T are the intersection points where the rays meet the two circles. There are two points where the common circle of these two circles intersects with these four points on the common circle. As a result, the two lines connecting P and S and Q and T cross at the radical centre of the three circles, lying on the radical axis of the circles. Similarly, the line joining the two anti-homologous points on distinct circles and their tangents forms an isosceles triangle with equal-length tangents. As a result, these tangents meet on the radical axis.

Algebraic construction

The distances between K and B and K and V are x1 and x2, respectively. The distances from J to B and V are d1 and d2.

The radical axis intersects the line segment joining the two circles’ centres B and V at K. So, x1 + x2 = D, the distance between B and V.

Consider a point J on the radical axis, with distances d1 and d2 to B and V. Due to d12 − r12 = d22 − r22, the Pythagorean theorem may describe distances in terms of x1, x2, and L, the distance from J to K.

L2 + x12 − r12 = L2 + x22 − r22

We can write the equation cancelling L2 on both sides.

x12 – x22 = r12 − r22

Divide both sides by D = x1 + x2 to get x1 − x2 = r12 − r22/D.

Adding this to x1 + x2 = D gives formula for x1,  2x1 = D + r12 − r22/D

Subtracting the same equation gives 2x2 = D − r12− r22/D.

Conclusion

The point where tangents are the same is the radical axis of two circles. Any circle that divides both circles orthogonally has its radical axis in the centre. On the radical axis, a point P has the same power as a point P on any of the two circles, r1 and r2. Straight lines perpendicular to the lines connecting the circles’ centres are radical axis. The radical axis connects two circles’ points of intersection of two circles. Any circle on the radical axis that passes between the two foci intersects the two circles orthogonally. So two radii of such a circle tangent both circles, satisfying the radical axis definition. by drawing circles with the same radical axis and centring them on the same line to form a coaxial circle pencil.