Question

Prove that if \(A\) and \(B\) are lattice points such that there are no lattice points on \(\overline{A B}\) and \(C\) is a lattice point that minimizes the distance to \(\overleftrightarrow{A B},\) then \(\triangle A B C\) is fundamental.

Short answer

Given conditions ensure \(\triangle ABC\) is fundamental by definition of unbroken hypotenuse and perpendicular lattice alignment.

Step-by-step solution

Understand Lattice Points and Fundamental Triangle

Lattice points are points with integer coordinates. A fundamental triangle is a right-angled triangle whose hypotenuse lies on the line connecting lattice points but has no other lattice points on it.

Condition on Line Segment and Distance

Given that there are no lattice points on the segment \(\overline{A B}\) except \(A\) and \(B\), it implies that \(A\) and \(B\) are consecutive lattice points on the line defined by \(\overleftrightarrow{AB}\). The perpendicular from any point \(C\) to this line will not intersect other lattice points apart from \(C\) if it minimizes the distance to the line.

Minimizing Distance to the Line

A lattice point \(C\) minimizing the distance to \(\overleftrightarrow{A B}\) means \(C\) is directly 'below' or 'above' the line to minimize perpendicular distance, depending on the slope of \(\overleftrightarrow{A B}\). This places \(C\) at the closest integer coordinate to the line.

Establishing Triangle Fundamentals

When \(C\) is chosen as described, \(\triangle ABC\) is aligned such that it forms a right angle at \(C\). Since there are no points on \(\overline{A B}\) and \(C\) is integer aligned minimizing the distance, \(\triangle ABC\) is fundamental because its hypotenuse is the unobstructed segment \(\overline{AB}\) according to the lattice definition.

Conclusion

Since the hypotenuse, \(\overline{AB}\), satisfies the condition of having no other lattice points, and \(C\) creates a perpendicular height from a close integer point, \(\triangle ABC\) is ensured to be fundamental in the lattice context.

Key concepts

Fundamental Triangle

A fundamental triangle is a special type of right-angled triangle with unique properties based on lattice points. Lattice points are those which have both coordinates as integers, such as \( (x, y) \) where both \(x\) and \ y \ are integers. In a fundamental triangle, the hypotenuse joins two lattice points and there are no other lattice points along that segment. This absence of additional lattice points ensures that the triangle satisfies the criteria for being fundamental.

A fundamental triangle guarantees that it contains crucial geometric properties. These properties help simplify calculations in coordinate geometry, as you focus only on integer-based values. It also means that every side except the hypotenuse is aligned with the axes in some way, aiding in solving problems related to triangles with lattice points.

Right-Angled Triangle

Right-angled triangles are triangles with one of their angles measuring 90 degrees. This 90-degree angle is also called a right angle. When working with lattice points and forming triangles, identifying a right-angled triangle can simplify understanding of its properties and relations.

In the context of lattice geometry, a right-angled triangle takes a special role. Its sides often correspond to horizontal, vertical, or diagonal lines between lattice points. This alignment makes it easier to calculate side lengths using the Pythagorean theorem. For example:

• If two sides are \(a\) and \ b\ (formed by differences in integer coordinates), the hypotenuse can be described as \(\sqrt{a^2 + b^2}\).

Being right-angled, \ \triangle ABC \ readily fits into this structure when constructed using lattice points.

Perpendicular Distance

Perpendicular distance is the shortest distance from a point to a line, achieved by drawing a perpendicular line (or a right angle) between the point and the given line. In lattice geometry, determining this distance can help locate significant points such as point \( C \) in the triangle \ \triangle ABC \.

For any line defined between two points \( A \) and \ B \, ensuring that point \ C \ is at a perpendicular distance means that \ C \ lies directly "above" or "below" the line. This position helps minimize the distance between \ C \ and line \ \overleftrightarrow{AB} \. Using the line equation \ y = mx + c \, the perpendicular distance \ d \ from a point \ (x_1, y_1) \ to the line is calculated as:

• \ \frac{|mx_1 - y_1 + c|}{\sqrt{m^2 + 1}} \

For a lattice-based exercise, ensure \ C \'s coordinates align with integer values, fulfilling lattice point requirements.

Integer Coordinates

Integer coordinates refer to coordinates composed solely of integer values. They hold significant importance in discussions about lattice points because they define the exact placement of these points on a grid or plane. Understanding integer coordinates is crucial when solving geometry problems involving lattice points.

When both \(x\) and \(y\) in a coordinate pair are integers, the point lies directly on the grid lines of a standard Cartesian plane. This makes calculations regarding distances, slopes, and triangle properties more straightforward.

• Lattice points with integer coordinates simplify the determination of geometric constructions like fundamental triangles because sides often align with axes or diagonals.

In solving tasks that involve integer coordinates for triangles and other figures, all distances and angles can be computed using basic arithmetic and integer operations, enhancing precision and ease of understanding.