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Let $S$ be a simply connected Riemann surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{R}^{2}$, the points of half area of a triangle is a straight line)

Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{R}^{2}$, the points of half area of a triangle is a straight line)

Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.

Let $S$ be a simply connected Riemann surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{R}^{2}, the points of half area of a triangle is a straight line)

Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.

Let $S$ be a simply connected Riemann surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{R}^{2}$, the points of half area of a triangle is a straight line)

Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.

Let $S$ be a simply connected Riemann surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.

Let $S$ be a simply connected Riemann surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{R}^{2}, the points of half area of a triangle is a straight line)

Assume that for every (small) triangle, the points of half area is a geodesic.

What can be said about the geometry of $S$?(Its curvature)

Note By small triangle I mean "each point of $S$ has a neighborhood $U$ such that all triangles in $U$ satify the above property.