"Euclid Alone", a novelette by William F. Orr; first published in the 1975 anthology Orbit 16 (Damon Knight, ed.), available for borrowing (free but registration required) from the Internet Archive; reprinted in the 1987 anthology Mathenauts: Tales of Mathematical Wonder (Rudy Rucker, ed.), also available for borrowing from the Internet Archive.
Summary from the Mathematical Fiction site:
An administrator in the math department of a major research institute has to decide how to handle a paper which proves the inconsistency of Euclidean geometry.
There was an intricate structure made entirely of straight lines in a plane that intersected in named points and formed identifiable triangles, all related to one another by a carefully chosen pattern of congruences, similarities, and equal sides and angles.
This structure had been built very carefully to its present state by a process of repetitive partial construction. All the way along the freeway he had occupied himself with the task of mentally rebuilding the structure which he had spent almost the whole night examining. He would begin each time at the same starting point, building one line at a time, noting each label, each equality and similarity, until the structure reached a degree of complexity, as it did each time, that exceeded his ability to assimilate new information and which became manifest in the sudden complete loss of a necessary fact. At this point the only possibility was to begin again at the foundation. Each time, he got a little farther in the proof, and while it might seem at first a most inefficient method of construction, it would eventually result, not only in a completed proof, but also, and more importantly, in a complete intuitive familiarity with that proof, both in its overall conception and in all its particulars.
He sat frozen behind his desk. He had to talk to Hudson and stop today's mailing, to recall any copies that had already been sent out, to hold them until someone could make a final decision. Someone. He should run a check on the computer, and a projection too. Ordering these things in his mind was a difficult task. There were too many factors to tell what to do first. The whole pattern of his schedule was torn, and he had left his damn briefcase with that damn paper in the car. Construct angle F'G'H = angle GG'B. Then, if AJ is dropped perpendicular to BG from A, BJ = AJ and BG = F'H. Thus triangle ABG is congruent to . .. is congruent to . . . He rose, rushed to the blackboard, and began drawing furiously, attacking that hideous proof directly, headlong. He must find a fallacy. It must be false. He drew in three colors of chalk, erasing and redrawing segments in new proportions, stepping back across the room to view his diagram from a distance, making quick notations on the back of the piece of yellow paper on his desk, pacing the room jerkily and returning to the board to scowl, erase, and redraw. He hardly noticed it when Ruth buzzed to tell him Hudson was on the line. He strode to the desk, one eye on the board, and surprised himself by his handling of the situation.
"My schedule is terribly busy, Dr. Lucas, and I don't see how I can fit in another appointment—unless you will tell me the nature of the problem."
Lucas was not ready for this, not ready to reveal to anyone else the secret that, as far as he knew, he alone shared with Professor Paul David. He had not thought this far. He would have to tell another man the horrible thing that had been discovered, the horrible thing that had lain in wait for discovery all these centuries. His face covered with sweat, his hand sticky against the plastic receiver, he controlled his voice as much as he could and said: "A disproof of Euclid, sir. One of our fundees has produced a proof of the inconsistency of Euclid . . . that Euclid is not true, cannot be true . . ."