Compiled and Solved Problems in Geometry and Trigonometry

Florentin Smarandache ‖đ?‘€đ??¸â€– ‖đ?‘€đ??šâ€–

= đ?‘˜. Let M' be another point with the same property, namely

‖đ?‘€â€˛đ??¸â€˛â€– ‖đ?‘€â€˛đ??šâ€˛â€–

= đ?‘˜.

đ?‘ƒ, đ?‘€, đ?‘€â€˛ collinear â&#x;š the locus is a line that passes through đ?‘ƒ.

When the points are in âˆ˘đ??śđ?‘ƒđ??ľ we obtain one more line that passes through đ?‘ƒ.

Thus the locus is formed by two concurrent lines through đ?‘ƒ, from which we eliminate point đ?‘ƒ, because the distances from đ?‘ƒ to both lines are 0 and their ratio is indefinite.

Vice versa, if points đ?‘ and đ?‘ ′ are on the same line passing through đ?‘ƒ, the ratio of their distances to lines đ??´đ??ľ and đ??śđ??ˇ is constant.

Solution to Problem 37.

We show in the same way as in the previous problem that: ‖đ?‘€đ??¸â€– ‖đ?‘€đ??¸â€– đ?‘˜ đ?‘˜đ?‘‘ =đ?‘˜â&#x;š = â&#x;š ‖đ?‘€đ??¸â€– = , ‖đ?‘€đ??šâ€– ‖đ?‘€đ??šâ€– + ‖đ?‘€đ??¸â€– 1 − đ?‘˜ 1+đ?‘˜ and the locus of the points which are located at a constant distance from a given line is a parallel to the respective line, located between the two parallels.

If ||đ??´đ??ľ|| > ||đ??śđ??ˇ|| â&#x;š đ?&#x2018;&#x2018;(đ?&#x2018;&#x20AC;đ??´đ??¸) < đ?&#x2018;&#x2018;(đ?&#x2018;&#x20AC;đ??śđ??ˇ).

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