Vance's goal is to construct a circle that is tangent to all three sides of an acute scalene triangle LMN. His plan involves drawing altitudes, finding their intersection point, and then using this point as the center of the circle. While the initial steps are correct, there's a crucial flaw in his final step that need...
Vance's Construction Steps
Let's break down Vance's construction steps:
- Draw altitudes from vertex L and vertex M: This step is essential as it creates lines perpendicular to the opposite sides of the triangle. These lines will intersect at a point known as the orthocenter of the triangle.
- Mark their intersection point as O: The intersection point, O, represents the orthocenter of the triangle LMN.
- Draw the perpendicular from point O to side MN and mark the intersection point as P: This step is significant because it establishes a point P that is equidistant from the center of the circle (point O) and the side MN.
- Draw the circle centered at point O which will pass through point P: This is where Vance's plan falters. While the circle will indeed pass through point P, it might not be tangent to all three sides of the triangle.
The Flaw in Vance's Plan
The flaw lies in the assumption that the circle centered at point O and passing through point P will automatically be tangent to all three sides of the triangle. This is not always true. To understand why, let's visualize the scenario:
Consider the acute scalene triangle LMN. The orthocenter O lies within the triangle. When we draw a perpendicular from O to side MN, we get point P. The circle centered at O with radius OP will indeed pass through point P. However, this circle might not be tangent to sides LM and LN.
Imagine a scenario where the circle intersects sides LM and LN at two distinct points. In this case, the circle wouldn't be tangent to these sides, as it would have two points of contact instead of one.
The Correct Approach: The Incenter
To construct a circle tangent to all three sides of the triangle, Vance needs to focus on the incenter of the triangle. The incenter is the point where the angle bisectors of the triangle intersect. Here's how to find the incenter and construct the desired circle:
- Draw the angle bisectors of triangle LMN: An angle bisector divides an angle into two equal angles. Draw the angle bisectors of angles L, M, and N.
- Mark the intersection point of the angle bisectors as I: This intersection point, I, represents the incenter of the triangle.
- Draw the perpendicular from point I to any side of the triangle: This perpendicular will be the radius of the incircle.
- Draw the circle centered at point I with the calculated radius: This circle will be tangent to all three sides of the triangle.
Why the Incenter Works
The incenter holds a unique property: it is equidistant from all three sides of the triangle. This property makes it the ideal center for the incircle, the circle inscribed within the triangle and tangent to all its sides.
Conclusion
Vance's initial plan was a good start but needed revision. While the orthocenter plays a vital role in various geometric constructions, it's not the key to constructing a circle tangent to all three sides of a triangle. Instead, focusing on the incenter, the point where the angle bisectors intersect, ensures the construction of the desired circle.
Additional Considerations
While the incenter allows for the construction of the incircle, it's important to note that the existence and properties of the incircle are influenced by the nature of the triangle. For instance:
- Acute Triangles: All acute triangles have an incircle.
- Obtuse Triangles: Obtuse triangles also have an incircle, but the incenter lies outside the triangle.
- Right Triangles: In right triangles, the incenter coincides with the midpoint of the hypotenuse.
Understanding these nuances is crucial for applying the concept of the incenter effectively in various geometric scenarios.