circles P and Q are externally tangent at T. Horizontal lines l and m are, respectively, tangent to circles P and Q at R and S. Show that RT must contain S.
To show that RT must contain S, we can prove that the line TS is parallel to the line RT by using the tangent properties.
Here's how we can approach the problem step-by-step:
Step 1: Draw the diagram.
Draw two circles, P and Q, that are externally tangent at point T. Draw horizontal lines l and m that are tangent to circles P and Q at points R and S, respectively.
Step 2: Identify the given information.
According to the problem, lines l and m are tangent to circles P and Q at points R and S. We need to show that line RT must contain point S.
Step 3: Observe the tangent properties.
When a line is tangent to a circle, it is perpendicular to the radius of the circle drawn to the point of tangency.
Step 4: Analyze the tangent properties in the given diagram.
Since line l is tangent to circle P at point R, line l is perpendicular to the radius of circle P drawn to point R. Similarly, since line m is tangent to circle Q at point S, line m is perpendicular to the radius of circle Q drawn to point S.
Step 5: Apply the properties.
If two lines are perpendicular to the same line, they are parallel to each other. In our case, line l is perpendicular to the radius of circle P drawn to point R, and line m is perpendicular to the radius of circle Q drawn to point S. Since both lines l and m are perpendicular to their respective radii, they are parallel to each other.
Step 6: Use parallel lines property.
If two lines are parallel, any transversal line that intersects one of them will also intersect the other. In our case, line RT is a transversal line that intersects line l at point R. Therefore, according to the property mentioned above, line RT must also intersect line m at point S.
Step 7: Conclude the proof.
Since we have shown that line RT intersects line m at point S, we can conclude that RT must contain S.
By using the above reasoning and the tangent properties of the circles, we have demonstrated that RT must contain S.