Congruent Chords in the Same Circle are Equidistant from the Center

Two congruent chords of a circle are equidistant from its center.
NOTE: Two chords are said to be congruent if they are equal in length.


congruent-chord-circle

Consider the equation of the circle

{x^2} + {y^2} = {r^2}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Suppose that AB and CD are congruent chords with A\left( {{x_1},{y_1}} \right), B\left( {{x_2},{y_2}} \right), C\left( {{x_3},{y_3}} \right) and D\left( {{x_4},{y_4}} \right) as shown in the given diagram. Since the circle passes through the points A,\,B,\,C and D, the equation of the circle becomes

\begin{gathered} {x_1}^2 + {y_1}^2 = {r^2}\,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right) \\ {x_2}^2 + {y_2}^2 = {r^2}\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right) \\ {x_3}^2 + {y_3}^2 = {r^2}\,\,\,\,{\text{ - - - }}\left( {{\text{iv}}} \right) \\ {x_4}^2 + {y_4}^2 = {r^2}\,\,\,\,{\text{ - - - }}\left( {\text{v}} \right) \\ \end{gathered}

Since M is the midpoint of the chord AB, so

M\left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)

Since N is the midpoint of the chord CD, so

N\left( {\frac{{{x_3} + {x_4}}}{2},\frac{{{y_3} + {y_4}}}{2}} \right)

Now we shall find the distance between O and M, as follows:

\begin{gathered} {\left| {OM} \right|^2} = {\left( {\frac{{{x_1} + {x_2}}}{2} - 0} \right)^2} + {\left( {\frac{{{y_1} + {y_2}}}{2} - 0} \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {\frac{{{x_1} + {x_2}}}{2}} \right)^2} + {\left( {\frac{{{y_1} + {y_2}}}{2}} \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\left( {{x_1} + {x_2}} \right)}^2}}}{4} + \frac{{{{\left( {{y_1} + {y_2}} \right)}^2}}}{4} = \frac{{{{\left( {{x_1} + {x_2}} \right)}^2} + {{\left( {{y_1} + {y_2}} \right)}^2}}}{4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{x_1}^2 + {x_2}^2 + 2{x_1}{x_2} + {y_1}^2 + {y_2}^2 + 2{y_1}{y_2}}}{4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{x_1}^2 + {y_1}^2 + {x_2}^2 + {y_2}^2 + 2{x_1}{x_2} + 2{y_1}{y_2}}}{4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{r^2} + {r^2} + 2{x_1}{x_2} + 2{y_1}{y_2}}}{4} = \frac{{2\left( {{r^2} + {x_1}{x_2} + {y_1}{y_2}} \right)}}{4} \\ \Rightarrow {\left| {OM} \right|^2} = \frac{{{r^2} + {x_1}{x_2} + {y_1}{y_2}}}{2}\,\,\,\,{\text{ - - - }}\left( {{\text{vi}}} \right) \\ \end{gathered}

Similarly, we can show that

 \Rightarrow {\left| {ON} \right|^2} = \frac{{{r^2} + {x_3}{x_4} + {y_3}{y_4}}}{2}\,\,\,\,{\text{ - - - }}\left( {{\text{vii}}} \right)

Since AB and CD are congruent chords,  {\left| {AB} \right|^2} = {\left| {CD} \right|^2}

\begin{gathered} \Rightarrow {\left( {{x_2} - {x_1}} \right)^2} + {\left( {{y_2} - {y_1}} \right)^2} = {\left( {{x_4} - {x_3}} \right)^2} + {\left( {{y_4} - {y_3}} \right)^2} \\ \Rightarrow {x_1}^2 + {x_2}^2 - 2{x_1}{x_2} + {y_1}^2 + {y_2}^2 - 2{y_1}{y_2} = \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_3}^2 + {x_4}^2 - 2{x_3}{x_4} + {y_3}^2 + {y_4}^2 - 2{y_3}{y_4} \\ \end{gathered}

Using equations (2) and (3), we get the following result:

\begin{gathered} \Rightarrow {r^2} + {r^2} - 2{x_1}{x_2} - 2{y_1}{y_2} = {r^2} + {r^2} - 2{x_3}{x_4} - 2{y_3}{y_4} \\ \Rightarrow - 2{x_1}{x_2} - 2{y_1}{y_2} = - 2{x_3}{x_4} - 2{y_3}{y_4} \\ \Rightarrow {x_1}{x_2} + {y_1}{y_2} = {x_3}{x_4} + {y_3}{y_4} \\ \end{gathered}

Adding {r^2} to both sides of the above equation, we have

\begin{gathered} \Rightarrow {r^2} + {x_1}{x_2} + {y_1}{y_2} = {r^2} + {x_3}{x_4} + {y_3}{y_4} \\ \Rightarrow \frac{{{r^2} + {x_1}{x_2} + {y_1}{y_2}}}{2} = \frac{{{r^2} + {x_3}{x_4} + {y_3}{y_4}}}{2}\,\,\,{\text{ - - - }}\left( {{\text{viii}}} \right) \\ \end{gathered}

Using equations (vi) and (vii) in equation (viii), we get

\begin{gathered} {\left| {OM} \right|^2} = {\left| {ON} \right|^2} \\ \Rightarrow \left| {OM} \right| = \left| {ON} \right| \\ \end{gathered}

This shows that the congruent chords of a circle are equidistant from its center.