Question

determining where the points of concurrency lie
where do the points of concurrency of certain types of triangles lie?
the circumcenter of an obtuse triangle lies the triangle.
the incenter of a right triangle lies the triangle.
the point of concurrency of the angle bisectors of an acute triangle lies the triangle.
the point of concurrency that is equidistant from the vertices of a right triangle lies the triangle.

Explanation:

To solve this, we recall properties of points of concurrency in triangles:

1. Circumcenter of an obtuse triangle

The circumcenter is the intersection of perpendicular bisectors. For an obtuse triangle, it lies outside the triangle.

2. Incenter of a right triangle

The incenter is the intersection of angle bisectors (and center of the incircle). For any triangle (including right triangles), the incenter lies inside the triangle.

3. Point of concurrency of angle bisectors (incenter) of an acute triangle

The incenter (intersection of angle bisectors) of any triangle, including acute triangles, lies inside the triangle.

4. Point equidistant from vertices (circumcenter) of a right triangle

The circumcenter of a right triangle is at the midpoint of the hypotenuse, which lies on the triangle (specifically, on the hypotenuse).

Final Answers:

1. The circumcenter of an obtuse triangle lies $\boldsymbol{\text{outside}}$ the triangle.
2. The incenter of a right triangle lies $\boldsymbol{\text{inside}}$ the triangle.
3. The point of concurrency of the angle bisectors of an acute triangle lies $\boldsymbol{\text{inside}}$ the triangle.
4. The point of concurrency that is equidistant from the vertices of a right triangle lies $\boldsymbol{\text{on}}$ the triangle.

Answer:

To solve this, we recall properties of points of concurrency in triangles:

1. Circumcenter of an obtuse triangle

The circumcenter is the intersection of perpendicular bisectors. For an obtuse triangle, it lies outside the triangle.

2. Incenter of a right triangle

The incenter is the intersection of angle bisectors (and center of the incircle). For any triangle (including right triangles), the incenter lies inside the triangle.

3. Point of concurrency of angle bisectors (incenter) of an acute triangle

The incenter (intersection of angle bisectors) of any triangle, including acute triangles, lies inside the triangle.

4. Point equidistant from vertices (circumcenter) of a right triangle

The circumcenter of a right triangle is at the midpoint of the hypotenuse, which lies on the triangle (specifically, on the hypotenuse).

Final Answers:

1. The circumcenter of an obtuse triangle lies $\boldsymbol{\text{outside}}$ the triangle.
2. The incenter of a right triangle lies $\boldsymbol{\text{inside}}$ the triangle.
3. The point of concurrency of the angle bisectors of an acute triangle lies $\boldsymbol{\text{inside}}$ the triangle.
4. The point of concurrency that is equidistant from the vertices of a right triangle lies $\boldsymbol{\text{on}}$ the triangle.